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LIBRARY 


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UNIVERSITY  OF   CALIFORNIA 

LIBRARY 

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DEPARTMENT   OF   PHYSICS 

KM NQV.14iim 

Accessions  No. 6.«L.<7....        Book  No... / 


SCIENTIFIC  MEMOIRS 

EDITED  BY 

J.  S.  AMES,  PH.D. 

PROFESSOR    OF    PHYSICS    IN    JOHNS    HOPKINS    UNIVERSITY 


IX. 
THE  LAWS  OF  GRAVITATION 


I 


THE 


LAWS   OF   GRAVITATION 


MEMOIKS    BY    NEWTON,    BOUGUEK 
AND    CAVENDISH 

TOGETHER  WITH  ABSTRACTS  OF  OTHER 
IMPORTANT  MEMOIRS 


TRANSLATED   AND  EDITED   BY 

A.  STANLEY  MACKENZIE,  PH.D. 

PROFRSSOR  OF  PHYSICS  IN  BRYN  MAWR  COLLEGE 


NEW   YORK   •:•   CINCINNATI   •:•   CHICAGO 

AMERICAN    BOOK    COMPANY 


Copyright,  1900,  by  AMERICAN  BOOK  COMPANY. 
w.  P.  i 


222544 


GENERAL    CONTENTS 


PAGE 

Preface v 

History  of  the  subject  before  the  appearance  of  Newton's  Pnncipia.  1 

Extracts  from  Newton's  Principia  and  System  of  the  World 9 

Biographical  sketch  of  Newton 19 

Bouguer's  The  Figure  of  the  Earth 23 

Biographical  sketch  of  Bouguer 44 

The  Bertier  controversy 47 

Account  of  Maskelyne's  experiments  on  Schehallien 53 

Cavendish's  Experiments  to  determine  the  mean  density  of  the  Earth ...  59 

Biographical  sketch  of  Cavendish 107 

Historical  account  of  the  experiments  made  since  the  time  of  Caven- 
dish   Ill 

Table  of  results  of  experiments 143 

Bibliography 145 

Index..                                                                            157 


PREFACE 


IN  preparing  this  volume,  the  ninth  in  the  Scientific  Me- 
moirs series,  the  editor  has  had  in  mind  the  fact  that  the  most 
important  of  the  memoirs  here  dealt  with,  that  of  Cavendish, 
is  frequently  given  for  detailed  study  to  young  physicists  in 
order  to  train  them  in  the  art  of  reading  for  themselves  period- 
ical scientific  literature.  Certainly  no  better  piece  of  work 
could  be  used  for  the  purpose,  whether  one  considers  the 
intrinsic  importance  of  the  subject-matter,  the  keenness  of 
argument  and  the  logical  presentation  in  detail,  or  the  use  and 
design  of  apparatus  and  the  treatment  of  sources  of  error. 
The  main  objections  to  Cavendish's  work  are  those  he  himself 
pointed  out,  and  it  is  important  to  notice  that,  notwithstand- 
ing all  the  advance  in  the  refinement  and  manipulation  of 
apparatus  which  has  been  made  during  the  century  that  has 
elapsed  since  the  date  of  Cavendish's  experiment,  his  value  for 
the  mean  specific  gravity  of  the  earth,  5.448,  must  still  be  con- 
sidered one  of  the  most  reliable,  being  not  far  from  the  latest 
results  of  Poynting,  Konig  and  Richarz  and  Krigar-Menzel, 
Boys  and  Braun. 

Believing  that  we  in  America  devote  insufficient  time,  if 
any,  to  a  study  of  Newton's  great  work,  the  editor  has  thought 
it  well  to  incorporate  with  the  memoirs  on  the  experimental 
investigation  of  gravitational  attraction  the  statements  of  New- 
ton himself  concerning  that  subject. 

The  laws  of  gravitation  are  embodied  in  the  formula, 
,    r  mm' 
f=(*~f 

which  says  that  the  attraction  between  two  particles  of  matter 
is  directly  proportional  to  the  product  of  their  masses,  inverse- 
ly proportional  to  the  square  of  the  distance  between  them, 
and  independent  of  the  kind  of  matter  and  of  the  intervening 


PREFACE 

medium.  G  is  then  a  constant  in  nature,  the  Gravitation  Con- 
stant. It  is  more  common  perhaps  to  speak  of  the  law,  than 
of  the  laws,  of  gravitation  ;  this  has  no  doubt  arisen  from  the 
fact  that  they  can  be  stated  in  a  single  mathematical  formula. 
The  best  evidence  of  the  truth  of  these  laws  is  indirect,  for, 
assuming  them  valid,  astronomical  measurements  show  that 
they  account  for  all  the  motions  of  the  heavenly  bodies.  Such 
measurements  do  not,  however,  enable  us  to  find  the  numer- 
ical value  of  G;  for  that  purpose  we  must  determine  the  at- 
traction between  two  masses  of  known  amount  at  a  known 
distance  apart.  It  is  with  experiments  of  this  character  that 
the  present  volume  has  to  deal.  As  the  masses  used  in  such 
experiments  vary  from  a  metal  sphere  of  a  few  tenths  of  an 
inch  in  diameter  to  a  huge  mountain  mass,  or  to  a  shell  of  the 
earth's  crust  1250  ft.  in  thickness,  and  as  the  attraction  has 
been  observed  with  such  different  instruments  as  the  plumb- 
line,  the  pendulum,  the  torsion  balance,  the  pendulum  bal- 
ance and  the  beam  balance,  and  yet  the  resulting  value  of  G  is 
always  about  the  same,  we  can  regard  these  experiments  as 
constituting  a  further  proof  of  Newton's  laws,  and  the  editor 
has  accordingly  felt  justified  in  using  the  title  given.  As- 
suming the  earth  to  be  a  sphere,  the  value  of  G  is  connected 
with  the  value  of  the  mean  specific  gravity  of  the  earth,  A, 
by  the  equation 


where  g  Is  the  acceleration  due  to  gravity,  and  R  the  radius  of 
the  earth  ;  and,  accordingly,  it  is  quite  usual  to  state  that  the 
aim  of  the  above  experiments  is  to  find  the  mean  density  of 
the  earth. 

The  work  on  the  attraction  of  mountain  masses  by  the 
French  Academicians  Bouguer  and  de  la  Condamine  in  Peru  is 
of  very  great  importance,  and  is  not  known  as  it  deserves  to 
be;  almost  all  of  their  account  of  the  work  is  therefore  here 
presented.  It  will  be  seen  that  they  were  the  pioneers  in  two 
of  the  methods  which  have  been  used  for  the  measurement  of 
gravitational  attraction  ;  and  although,  on  account  of  imper- 
fect instruments  and  unfavourable  local  conditions,  their  nu- 
merical results  are  untrustworthy,  they  give  the  theory  and 
method  of  the  experiments  with  great  originality  and  clear- 
ness. Such  notes  have  been  added  to  the  memoirs  as  seemed 

vi 


PREFACE 

necessary  to  prevent  the  reader  from  wasting  time  over  obscure 
and  inaccurate  passages,  and  to  suggest  material  for  collateral 
reading. 

An  eifort  has  been  made  to  present  along  with  the  memoirs 
a  brief  historical  account  of  the  various  modes  of  experiment 
used  for  finding  the  mean  specific  gravity  of  the  earth,  and  a 
table  of  results  is  added.  As  the  literature  on  the  subject 
before  the  present  century  is  not  always  easily  obtainable,  the 
treatment  of  the  matter  for  that  period  is  given  in  compara- 
tively greater  detail.  Believing  that  a  bibliography  contain- 
ing every  important  reference  to  the  subject  is  an  essential 
feature  of  a  work  of  this  kind,  the  editor  has  endeavoured  to 
make  himself  familiar  with  the  whole  of  the  very  extensive 
literature  relating  to  it,  and  accordingly  is  fairly  confident 
that  no  important  memoir  has  escaped  his  observation.  From 
the  mass  of  material  thus  collected  the  bibliography  given  at 
the  end  of  the  volume  has  been  compiled.  In  order  to  keep 
within  the  limits  of  space  assigned,  some  references  had  to 
be  omitted,  but  they  relate  mainly  to  recent  work,  and  it  is 
believed  that  they  contain  nothing  of  importance. 

No  effort  has  been  made  to  deal  with  the  mathematical  side 
of  the  subject ;  accordingly  the  memoirs  of  Laplace,  Legendre, 
Ivory,  etc.,  which  deal  with  the  finding  of  the  mean  specific 
gravity  of  the  earth  by  means  of  analytical  methods  are  not 
referred  to ;  but  it  is  hoped  that  all  the  more  important  ex- 
perimental investigations  have  been  touched  upon. 

A.  STANLEY  MACKENZIE. 

BKYN  MAWR,  October,  1899. 

vii 


HISTORY  OF  THE   SUBJECT 

BEFORE   THE 
APPEARANCE  OF  NEWTON'S  "PRINCIPIA" 


DR.  GILBERT'S  contributions  to  the  speculations  on  gravita- 
tion are  among  the  most  important  of  the  early  writings  on  that 
subject,  although  to  Kepler  also  must  credit  be  given  for  a 
deep  insight  into  its  nature;  the  latter  announces  in  his  intro- 
duction to  the  Astronomia  Nova,  published  in  16Q9,  his  belief 
in  the  perfect  reciprocity  of  the  action  of  gravitation,  and  in  its 
application  to  the  whole  material  universe.  Gilbert  was  led  by 
his  researches  on  magnetism  to  the  conclusion  that  the  force  of 
gravity  was  due  to  the  magnetic  properties  of  the  earth  ;  and 
in  1600  announced  [1*,  I,  21]  his  opinion  that  bodies  when  re- 
moved to  a  great  distance  from  the  earth  would  gradually  lose 
their  motion  downwards.  The  earliest  proposals  we  find  for 
investigating  whether  such  changes  occur  in  the  force  of  gravity 
are  in  the  works  of  Francis  Bacon  [2,  Nov.  Org.  II,  36,  and 
Hist.  Nat.  I,  33].  He  maintained  that  this  force  decreased 
both  inwards  and  outwards  from  the  surface  of  the  earth,  and 
suggested  experiments  to  test  his  views.  He  would  take  two 
clocks,  one  actuated  by  weights  and  the  other  by  the  compres- 
sion of  an  iron  spring,  and  regulate  them  so  that  they  would 
run  at  the  same  rate.  The  clock  actuated  by  weights  was  then 
to  be  placed  at  the  top  of  some  high  steeple,  and  at  the  bottom 
of  a  mine,  and  its  rate  at  each  place  compared  with  that  of  the 
other,  which  remained  at  the  surface.  There  is  no  record  .of 
any  trial  of  the  experiment  at  that  time. 

After  the  founding  of  the  Royal  Society  of  London  a  stimu- 
lus was  given  to  experimenting  upon  this  as  upon  many  other 

*  The  numbers  in  brackets  refer  to  the  Bibliography. 

A  1 


ON 

subjects.  /•&: |Kip£rJw,aS  \e&([  5be&>r<r that  society  by  Dr.  Power 
[10,  vol.  1,  p.  133]  on  December  3,  1662,  upon  "  Subterraneous 
Experiments."  A  pound  weight  and  68  yards  of  thread 
were  put  into  one  pan  of  a  scale  and  counterpoised.  The 
weight  was  then  lowered  into  a  pit  and  attached  by  means  of 
the  thread  to  the  scale-pan  held  directly  over  the  mouth  of  the 
pit ;  it  was  found  to  lose  in  weight  by  at  least  an  ounce.*  Three 
weeks  later  Hooke  [10,  vol.  1,  p.  163]  made  a  report  to  the 
society  on  some  experiments  he  had  performed  at  Westminster 
Abbey.  The  report  is  worth  reprinting,  as  giving  some  idea 
of  the  method  employed  in  such  experiments,  and  of  the  state 
of  knowledge  upon  the  subject  at  the  time  when  Newton  first 
took  it  up.  For  it  will  be  remembered  that  it  was  in  1665  that 
Newton  was  led  by  his  speculations  on  gravity  to  imagine  that 
since  this  action  did  not  sensibly  diminish  with  small  changes 
in  height,  it  might  perhaps  extend  to  the  moon,  and  be  the 
cause  of  that  body's  being  retained  in  her  orbit.  Pursuing  his 
train  of  thought,  he  extended  this  explanation  to  the  sun  and 
planets;  and,  taking  into  consideration  Kepler's  laws,  it  was 
therefore  necessary  that  the  force  must  fall  olf  in  the  inverse 
ratio  of  the  square  of  the  distance.  When  applying  this  law  of 
decrease  from  the  earth  to  the  moon,  Newton  used  in  deduc- 
ing the  length  of  the  radius  of  the  earth  the  rough  estimate, 
then  current,  of  60  miles  to  a  degree  of  latitude,  instead  of 
nearly  69^  ;  and  as  a  consequence  the  calculated  motion  of  the 
moon  did  not  agree  with  the  observed  motion.  He  thereupon 
laid  aside  for  the  time  being  any  further  thought  upon  the 
matter.  His  attention  was  again  called  to  the  subject  by  a 
letter  from  Hooke  in  1679,  and,  Picard  having  in  the  meantime 
measured  the  earth,  Newton  was  able  to  apply  the  correct  data 
to  the  problem  and  to  arrive  at  a  beautiful  agreement  of  the 
calculated  with  the  observed  behaviour  of  the  moon.  From 
that  time  date  the  wonderful  researches  which  were  the  founda- 
tion of  the  Principia.  The  following  is  Hooke's  report : 

"In  prosecution  of  my  Lord  Verulam's  experiment  concern- 
ing the  decrease  of  gravity,  the  farther  a  body  is  removed  be- 
low the  surface  of  the  earth,  I  made  trial,  whether  any  such 
difference  in  the  weight  of  bodies  could  be  found  by  their 

*  According  to  Le  Sage  [34],  Descartes  hud  suggested  a  similar  under- 
taking twenty-five  years  earlier  in  a  letter  to  Mcrsenne. 

2 


THE    LAWS    OF    GRAVITATION 

nearer  or  farther  removal  from  that  surface  upwards.  To  this 
end  I  took  a  pair  of  exact  scales  and  weights,  and  went  to  a 
convenient  place  upon  Westminster  Abbey,  where  was  a  per- 
pendicular height  above  the  leads  of  a  subjacent  building, 
which  by  measure  I  found  threescore  and  eleven  foot.  Here 
counterpoising  a  piece  of  iron  (which  weighed  about  15  ounces 
troy)  and  packthread  enough  to  reach  from  the  top  to  the  bot- 
tom, I  found  the  counterpoise  to  be  of  troy- weight  seventeen 
ounces  and  thirty  grains.  Then  letting  down  the  iron  by  the 
thread,  till  it  almost  touched  the  subjacent  leads,  I  tried  what 
alteration  there  had  happened  to  its  weight,  and  found,  that 
the  iron  preponderated  the  former  counterpoise  somewhat  more 
than  ten  grains.  Then  drawing  up  the  iron  and  thread  with 
all  the  diligence  possibly  I  could,  that  it  might  neither  get  nor 
lose  any  thing  by  touching  the  perpendicular  wall,  I  found  by 
putting  the  iron  and  packthread  again  into  its  scale,  that  it 
kept  its  last  equilibrium;  and  therefore  concluded,  that  it  had 
not  received  any  sensible  difference  of  weight  from  its  nearness 
to  or  distance  from  the  earth.  I  repeated  the  trial  in  the  same 
place,  but  found,  that  it  had  not  altered  its  equilibrium  (as  in 
the  first  trial)  neither  at  the  bottom,  nor  after  I  had  drawn  it 
up  again  ;  which  made  me  guess,  that  the  first  preponderating 
of  the  scale  was  from  the  moisture  of  the  air,  or  the  like,  that 
had  stuck  to  the  string,  and  so  made  it  heavier.  In  pursuance 
of  this  experiment,  I  removed  to  another  place  of  the  Abbey, 
that  was  just  the  same  distance  from  the  ground,  that  the  for- 
mer was  from  the  leads;  and  upon  repeating  the  trial  there 
with  the  former  diligence,  I  found  not  any  sensible  alteration 
of  the  equilibrium,  either  before  or  after  I  had  drawn  it  up; 
which  farther  confirmed  me,  that  the  first  alteration  proceeded 
from  some  other  accident,  and  not  from  the  differing  gravity  of 
the  same  body. 

"I  think  therefore  it  were  very  desirable,  from  the  determi- 
nation of  Dr.  Power's  trials,  wherein  he  found  such  difference 
of  weight,  that  it  were  examined  by  such  as  have  opportunity, 
first,  what  difference  there  is  in  the  density  and  pressure  of 
the  air,  and  what  of  that  condensation  of  gravity  may  be  as- 
cribed to  the  differing  degrees  of  heat  and  cold  at  the  top 
and  bottom,  which  may  be  easily  tried  with  a  common  weather- 
glass and  a  sealed-up  thermometer  ;  for  the  thermometer  will 
shew  what  of  the  change  is  to  be  ascribed  to  heat  and  cold, 

3 


MEMOIRS    ON 

and  the  weather-glass  will  shew  the  differing  condensation. 
Next,  for  the  knowing,  whether  this  alteration  of  gravity  pro- 
ceed from  the  density  and  gravity  of  the  ambient  air,  it  would 
be  requisite  to  make  use  of  some  very  light  body,  extended 
into  large  dimensions,  such  as  a  large  globe  of  glass  carefully 
stopt,  that  no  air  may  get  in  or  out;  for  if  the  alteration 
proceeded  from  the  magnetical  attraction  of  the  parts  of  the 
earth,  the  ball  will  lose  but  a  sixteenth  part  of  its  weight  (sup- 
posing a  lump  of  glass  held  the  same  proportion,  that  Dr. 
Power  found  in  brass);  but  if  it  proceed  from  the  density  of 
the  air,  it  may  lose  half,  or  perhaps  more.  Further,  it  were 
very  desireable,  that  the  current  of  the  air  in  that  place  were 
observed,  as  Sir  Robert  Moray  intimated  the  last  day.  Fourth- 
ly, I  think  it  were  worth  trial  to  counterpoise  a  light  and  heavy 
body  one  against  another  above,  and  to  carry  down  the  scales 
and  them  to  the  bottom,  and  observe  what  happens.  Fifthly, 
it  were  desireable,  that  trials  were  made,  by  the  letting  down  of 
other  both  heavier  and  lighter  bodies,  as  lead,  quicksilver,  gold, 
stones,  wood,  liquors,  animal  substances,  and  the  like.  Sixthly, 
it  were  to  be  wished,  that  trial  were  made  how  that  gravitation 
does  decrease  with  the  descent  of  the  body — that  is,  by  making 
trial,  how  much  the  body  grows  lighter  at  every  ten  or  twenty 
foot  distance.  These  trials,  if  accurately  made,  would  afford 
a  great  help  to  guess  at  the  cause  of  this  strange  phaenom- 
enon." 

Dr.  Power's  experiment  was  repeated  by  Dr.  Cotton,  and  an 
account  of  his  trials  was  given  to  the  Society  on  June  1,  1664 
[10,  vol.  1,  p.  433].  The  weight  was  J  lb.,  and  the  length  of  the 
string  36  yards.  A  loss  in  weight  of  £  oz.  was  fonnd. 

On  September  1,  1664  [6,  vol.  5,  p.  307],  we  find  a  reference 
to  some  experiments  made  at  St.  PauFs  Cathedral  by  a  com- 
mittee of  the  Royal  Society  consisting  of  Sir  R.  Moray,  Dr. 
Wilkins,  Dr.  Goddard,  Mr.  Palmer,  Mr.  Hill  and  Mr.  Hooke. 
The  results  of  these  experiments  were  given  to  the  Society  on 
September  14,  1664  [10,  vol.  1,  p.  466]  ;  the  weight  was  15  Ibs. 
troy,  the  string  about  200  ft.  long,  and  the  loss  of  weight  1 
drachm.  In  a  letter  to  Mr.  Boyle  [6,  vol.  5,  p.  536],  dated 
September  15th,  Mr.  Hooke  gives  more  details,  and  remarks 
that  the  balance  was  sensitive  enough  to  be  turned  by  a  few 
grains.  He  suggests  the  variation  of  the  density  of  the  air  as 
the  cause  of  the  loss  in  weight.  Boyle  [10,  vol.  1,  p.  470]  pro- 

4 


THE    LAWS    OF    GRAVITATION 

posed  that  Hooke's  suggestion  be  tested  by  making  the  sus- 
pended weight  of  a  large  glass  ball  loaded  with  mercury. 

At  a  meeting  of  the  Royal  Society  on  March  14,  1665,  Hooke 
reported  [10,  vol.  2,  p.  66,  and  6,  vol.  5,  p.  544]  that  he  had 
tried  Dr.  Power's  experiment  at  some  wells  near  Epsom  and 
had  found  no  loss  in  weight.  Similar  experiments  were  made 
by  Hooke  at  Ban  stead  Downs,  in  Surrey,  and  reported  on 
March  21,  1666  [10,  vol.  2,  p.  69,  and  6,  vol.  5,  pp.  355  and 
546].  The  string  was  330  ft.  long,  and  the  balance  sensitive 
to  a  grain,  yet  a  pound  shewed  no  change  in  weight  when  sus- 
pended at  the  bottom  of  the  well.  He  concludes  that  the 
power  of  gravity  cannot  be  magnetical,  as  Gilbert  had  sup- 
posed. He  says:  "But  in  truth  upon  the  consideration  of  the 
nature  of  the  theory,  we  may  find,  that  supposing  it  true,  that 
all  the  constituent  parts  of  the  earth  had  a  magnetical  power, 
the  decrease  of  gravity  would  be  almost  a  hundred  times  less 
than  a  grain  to  a  pound,  at  as  great  a  depth  as  fifty  fathom  ; 
for  if  we  consider  the  proportion  of  the  parts  of  the  earth 
placed  upon  one  side  beneath  the  stone,  with  the  parts  on  the 
other  side  above  it,  we  may  find  the  disproportion  greater. 
Unless  we  suppose  the  magnetism  of  the  parts  to  act  but  at  a 
very  little  distance,  which  I  think  the  experiments  made  in  the 
Abbey  and  St.  Paul's  will  not  allow  of.  If  therefore  there  be 
any  such  inequality  of  gravity,  we  must  have  some  ways  of 
trial  much  more  accurate  than  this  of  scales,  of  which  I  shall 
propound  two  sorts,"  etc.  It  is  interesting  to  notice  that  the 
considerations  upon  which  he  makes  his  computations  are  prac- 
tically those  used  by  Airy  in  his  Harton  Colliery  experiment. 

On  December  7,  1681  [10,  vol.  4,  p.  110],  Hooke  produced 
before  the  society  two  pendulum-clocks  adjusted  to  run  at  the 
same  rate.  He  proposed  to  put  one  at  the  top  and  the  other 
at  the  bottom  of  the  monument  on  Pish  Street  Hill,  and  ob- 
serve whether  they  would  keep  together.  No  notice  of  his 
having  tried  the  experiment  has  been  found.  This  is  the 
method  proposed  by  Bacon  and  used  by  Bouguer  and  many 
others. 

In  1682,  Hooke  read  before  the  Royal  Society  "A  Discourse 
of  the  Nature  of  Comets"  [4,  pp.  149-191],  in  which  he  gives 
his  ideas  on  the  subject  of  gravity  (particularly  on  pages  170- 
183).  He  considers  gravity  to  be  a  universal  principle,  in- 
herent in  all  matter,  propagated  by  the  same  medium  as  that 

5 


MEMOIRS    ON    THE    LAWS    OF    GRAVITATION 

by  means  of  which  light  is  conveyed,  with  unimaginable  celer- 
ity, to  indefinitely  great  distances,  and  with  a  power  varying 
with  the  distance.  He  sums  up  his  conceptions  on  gravitation 
in  nine  propositions,  which  are  of  great  interest,  in  that  they 
include  many  of  the  conceptions  of  Newton  on  this  subject, 
and  yet  were  published  four  years  before  the  Principia  ap- 
peared. 


PHILOSOPHIAE  NATURALIS  PRINCIPIA 
MATHEMATICA 

1st  Edition,  London,  1687.     2d  Edition,  Cambridge,  1713  (Cotes'  Edition). 
3d  Edition,  London,  1726  (Pemberton's  Edition) 

AND 

DE  MUNDI  SYSTEMATE 

London,  1727 
BY 

SIR  ISAAC   NEWTON 


(Extracts  taken  from  Davit's  Edition  of  Motte's  translation 
3  volumes,  London,  1803) 


CONTENTS 

PAGE 

On  the  attraction  of  spheres 9 

Law  of  the  distance 9 

Law  of  the  masses 13 

Variation  of  gravity  on  the  earth's  surface 14 

All  attraction  is  mutual 15 

Methods  of  showing  the  attraction  between  terrestrial  bodies 17 

Proof  of  its  existence . .  17 

Similar  discussion  for  the  case  of  celestial  bodies 18 

Final  statements  concerning  tJie  laws  of  gravitation 19 

8 


THE    MATHEMATICAL    PRINCIPLES    OF 
NATURAL  PHILOSOPHY 

AND 

SYSTEM    OF    THE    WORLD 

BY 

SIR    ISAAC    NEWTON 


BOOK  I.     PROPOSITION  LXX1V.     THEOREM  XXXIV. 

The  same  things  supposed  (if  to  the  several  points  of  a  given 
sphere  there  tend  equal  centripetal  forces  decreasing  in  a  du- 
plicate ratio  of  the  distances  from  the  points),  I  say,  that  a  cor- 
puscle situate  without  the,  sphere  is  attracted  with  a  force  recip- 
rocally proportional  to  the  square  of  its  distance  from  the 
centre. 

BOOK  I.     PROPOSITION  LXXV.     THEOREM  XXXV. 

If  to  the  several  points  of  a  given  sphere  there  tend  equal  cen- 
tripetal forces  decreasing  in  a  duplicate  ratio  of  the  distances 
from  the  points  ;  I  say,  that  another  similar  sphere  will  be  attract- 
ed by  it  with  a  force  reciprocally  proportional  to  the  square  of  the 
distance  of  the  centres. 

For  the  attraction  of  every  particle  is  reciprocally  as  the 
square  of  its  distance  from  the  centre  of  the  attracting  sphere 
(by  prop.  74),  and  is  therefore  the  same  as  if  that  whole  at- 
tracting force  issued  from  one  single  corpuscle  placed  in  the 
centre  of  this  sphere.  But  this  attraction  is  as  great  as  on  the 
other  hand  the  attraction  of  the  same  corpuscle  would  be,  if 
that  were  itself  attracted  by  the  several  particles  of  the  attract- 
ed sphere  with  the  same  force  with  which  they  are  attracted  by 

9 


MEMOIRS    ON 

it.  But  that  attraction  of  the  corpuscle  would  be  (by  prop.  74) 
reciprocally  proportional  to  the  square  of  its  distance  from  the 
centre  of  the  sphere  ;  therefore  the  attraction  of  the  sphere, 
equal  thereto,  is  also  in  the  same  ratio.  Q.  E.  D. 

Cor.  1.  The  attractions  of  spheres  towards  other  homogene- 
ous spheres  are  as  the  attracting  spheres  applied  to  the  squares 
of  the  distances  of  their  centres  from  the  centres  of  those 
which  they  attract. 

Cor.  2.  The  case  is  the  same  when  the  attracted  sphere  does 
also  attract.  For  the  several  points  of  the  one  attract  the  sev- 
eral points  of  the  other  with  the  same  force  with  which  they 
themselves  are  attracted  by  the  others  again  ;  and  therefore 
since  in  all  attractions  (by  law  3)  the  attracted  and  attracting 
point  are  both  equally  acted  on,  the  force  will  be  doubled  by 
their  mutual  attractions,  the  proportions  remaining. 

[Proposition  LXXVI.  proves  the  same  thing  for  spheres  made 
np  of  homogeneous  concentric  layers.] 

BOOK  III.     PROPOSITION  V.     THEOREM  V.     SCHOLIUM. 

The  force  which  retains  the  celestial  bodies  in  their  orbits 
has  been  hitherto  called  centripetal  force  ;  but  it  being  now 
made  plain  that  it  can  be  no  other  than  a  gravitating  force, 
we  shall  hereafter  call  it  gravity.  For  the  cause  of  that  cen- 
tripetal force  which  retains  the  moon  in  its  orbit  will  extend 
itself  to  all  the  planets. 

BOOK  III.     PROPOSITION  VI.     THEOREM  VI. 

That  all  bodies  gravitate  towards  every  planet ;  and  that  the 
weights  of  bodies  toiuards  any  the  same  planet,  at  equal  distances 
from  the  centre  of  the  planet,  are  proportional  to  the  quantities 
of  matter  which  they  severally  contain. 

It  has  been,  now  of  a  long  time,  observed  by  others,  that  all 
sorts  of  heavy  bodies  (allowance  being  made  for  the  inequality 
of  retardation  which  they  suffer  from  a  small  power  of  resist- 
ance in  the  air)  descend  to  the  earth  from  equal  heights  in 
equal  times  ;  and  that  equality  of  times  we.  may  distinguish  to 
a  great  accuracy,  by  the  help  of  pendulums.  I  tried  the  thing 
in  gold,  silver,  lead,  glass,  sand,  common  salt,  wood,  water, 
and  wheat.  »  I  provided  two  wooden  boxes,  round  and  equal ; 

10 


THE    LAWS    OF    GRAVITATION 

I  filled  the  one  with  wood,  and  suspended  an  equal  weight  of 
gold  (as  exactly  as  I  could)  in  the  centre  of  oscillation  of  the 
other.  The  boxes  hanging  by  equal  threads  of  11  feet  made  a 
couple  of  pendulums  perfectly  equal  in  weight  and  figure,  and 
equally  receiving  the  resistance  of  the  air.  And,  placing  the 
one  by  the  other,  I  observed  them  to  play  together  forwards 
and  backwards,  for  a  long  time,  with  equal  vibrations.  .  .  . 
and  the  like  happened  in  the  other  bodies.  By  these  experi- 
ments, in  bodies  of  the  same  weight,  I  could  manifestly  have 
discovered  a  difference  of  matter  less  than  the  thousandth  part 
of  the  whole,  had  any  such  been.  But,  without  all  doubt,  the 
nature  of  gravity  towards  the  planets  is  the  same  as  towards 
the  earth.  .  .  .  Moreover,  since  the  satellites  of  Jupiter  per- 
form their  revolutions  in  times  which  observe  the  sesquiplicate 
proportion  of  their  distances  from  Jupiter's  centre,  their  accel- 
erative  gravities  towards  Jupiter  will  be  reciprocally  as  the 
squares  of  their  distances  from  Jupiter's  centre — that  is,  equal 
at  equal  distances.  And,  therefore,  these  satellites,  if  sup- 
posed to  fall  towards  Jupiter  from  equal  heights,  would  describe 
equal  spaces  in  equal  times,  in  like  manner  as  heavy  bodies  do 
on  our  earth.  ...  If,  at  equal  distances  from  the  sun,  any  sat- 
ellite, in  proportion  to  the  quantity  of  its  matter,  did  gravitate 
towards  the  sun  with  a  force  greater  than  Jupiter  in  propor- 
tion to  his,  according  to  any  given  proportion,  suppose  of  d  to 
e  ;  then  the  distance  between  the  centres  of  the  sun  and  of  the 
satellite's  orbit  would  be  always  greater  than  the  distance, be- 
tween the  centres  of  the  sun  and  of  Jupiter  nearly  in  the  sub- 
duplicate  of  that  proportion  ;  as  by  some  computations  I  have 
found.  And  if  the  satellite  did  gravitate  to  wards  the  sun 
with  a  force,  lesser  in  the  proportion  of  e  to  d,,  the  distance  of 
the  centre  of  the  satellite's  orbit  from  the  sun  would  be  less 
than  the  distance  of  the  centre  of  Jupiter  from  the  sun  in  the 
subduplicate  of  the  same  proportion.  Therefore  if,  at  equal 
distances  from  the  sun,  the  accelerative  gravity  of  any  satell- 
ite towards  the  sun  were  greater  or  less  than  the  accelerative 
gravity  of  Jupiter  towards  the  sun  but  by  one  y^Vo  Parfc  °f  the 
whole  gravity,  the  distance  of  the  centre  of  the  satellite's  orbit 
from  the  sun  would  be  greater  or  less  than  the  distance  of  Ju- 
piter from  the  sun  by  one  -g-gVo  part  of  the  whole  distance — 
that  is,  by  a  fifth  part  of  the  distance  of  the  utmost  satellite 
from  the  centre  of  Jupiter  ;  an  eccentricity  of  the  orbit  which 

11 


MEMOIRS    ON 

would  be  very  sensible.  But  tbe  orbits  of  the  satellites  are 
concentric  to  Jupiter,  and  therefore  the  accelerative  gravities 
of  Jupiter,  and  of  all  its  satellites  towards  the  sun,  are  equal 
among  themselves.  .  .  . 

But  further  ;  the  weights  of  all  the  parts  of  every  planet 
towards  any  other  planet  are  one  to  another  as  the  matter  in 
the  several  parts  ;  for  if  some  parts  did  gravitate  more,  others 
less,  than  for  the  quantity  of  their  matter,  then  the  whole 
planet,  according  to  the  sort  of  parts  with  which  it  most 
abounds,  would  gravitate  more  or  less  than  in  proportion  to 
the  quantity  of  matter  in  the  whole.  Nor  is  it  of  any  moment 
whether  these  parts  are  external  or  internal  ;  for  if,  for  exam- 
ple, we  should  imagine  the  terrestrial  bodies  with  us  to  be 
raised  up  to  the  orb  of  the  moon,  to  be  there  compared  with 
its  body;  if  the  weights  of  such  bodies  were  to  the  weights  of 
the  external  parts  of  the  moon  as  the  quantities  of  matter  in 
the  one  and  in  the  other  respectively  ;  but  to  the  weights  of 
the  internal  parts  in  a  greater  or  less  proportion,  then  likewise 
the  weights  of  those  bodies  would  be  to  the  weight  of  the 
whole  moon  in  a  greater  or  less  proportion  ;  against  what  we 
have  shewed  above. 

Cor.  1.  Hence  the  weights  of  bodies  do  not  depend  upon 
their  forms  and  textures  ;  for  if  the  weights  could  be  altered 
with  the  forms,  they  would  be  greater  or  less,  according  to  the 
variety  of  forms,  in  equal  matter;  altogether  against  experience. 

Cor.  2.  Universally,  all  bodies  about  the  earth  gravitate 
towards  the  earth  ;  and  the  weights  of  all,  at  equal  distances 
from  the  earth's  centre,  are  as  the  quantities  of  matter  which 
they  severally  contain.  This  is  the  quality  of  all  bodies  within 
the  reach  of  our  experiments  ;  and  therefore  (by  rule  3)  to  be 
affirmed  of  all  bodies  whatsoever.  .  .  . 

Cor.  5.  The  power  of  gravity  is  of  a  different  nature  from  the 
power  of  magnetism  ;  for  the  magnetic  attraction  is  not  as  the 
matter  attracted.  Some  bodies  are  attracted  more  by  the 
magnet  ;  others  less  ;  most  bodies  not  at  all.  The  power  of 
magnetism  in  one  and  the  same  body  may  be  increased  and 
diminished  ;  and  is  sometimes  far  stronger,  for  the  quantity  of 
matter,  than  the  power  of  gravity  ;  and  in  receding  from  the 
magnet  decreases  not  in  the  duplicate  but  almost  in  the  tri- 
plicate proportion  of  the  distance,  as  nearly  as  I  could  judge 
from  some  rude  observations. 

12 


THE    L  A  W  S    0  F    GRAVITATION 

BOOK  III.     PROPOSITION  VII.     THEOREM  VII. 

That  there  is  a  power  of  gravity  tending  to  all  bodies,  pro- 
portioned to  the^everal  quantities  of  matter  which'  they  contain.. 

That  all  tbe/panets  mutually  gravitate  one  towards  another, 
we  have  proved  before  ;  as  well  as  that  the  force  of  gravity 
towards  every  one  of  them,  considered  apart,  is  reciprocally  as 
the  square  of  the  distancfe  of  places  from  the  centre  of  the 
planets  And  thence  (by  prop.  69,  book  I,  and  its  corollaries) 
it  follows,  that  the  gravity  tending  towards  all  the  planets  is 
proportional  to  the  matter  which  they  contain. 

Moreover,  since  all  the  parts  of  any  planet  A  gravitate  to- 
wards any  other  planet  B ;  and  the  gravity  of  every  part  is  to 
\\Q  gravity  of  the  whole  as  the  matter  of  the  part  to  the  matter 
tii  the  whole  ;  and  (by  law  3)  to  every  action  corresponds  an 
equal  reaction  ;  therefore  the  planet  B  will,  on  the  other  hand, 
gravitate  towards  all  the  parts  of  the  planet  A  ;  and  its  gravity 
towards  any  one  part  will  be  to  the  gravity  towards  the  whole 
as  the  matter  of  the  part  to  the  matter  of  the  whole.  Q.  E.  D. 

Cor.  1.  Therefore  the  force  of  gravity  towards  any  whole 
planet  arises  from,  and  is  compounded  of,  the  forces  of  gravity 
towards  all  its  parts.  Magnetic  and  electric  attractions  afford 
us  examples  of  this  ;  for  all  attraction  towards  the  whole  arises 
from  the  attractions  towards  the  several  parts.  The  thing  may 
be  easily  understood  in  gravity,  if  we  consider  a  greater  planet 
as  formed  of  a  number  of  lesser  planets  meeting  together  in 
one  globe  ;  for  hence,  it  would  appear  that  the  force  of  the  whole 
must  arise  from  the  forces  of  the  component  parts.  If  it  is 
objected  that,  according  to  this  law,  all  bodies  with  us  must 
mutually  gravitate  one  towards  another,  I  answer,  that  since 
the  gravitation  towards  these  bodies  is  to  the  gravitation  to- 
wards the  whole  earth  as  these  bodies  are  to  the  whole  earth, 
the  gravitation  towards  them  must  be  far  less  than  to  fall  under 
the  observation  of  our  senses. 

Cor.  2.  The  force  of  gravity  towards  the  several  equal  par- 
ticles of  any  body  is  reciprocally  as  the  square  of  the  distance 
of  places  from  the  particles  ;  as  appears  from  cor.  3,  prop.  74, 
book  I. 

[  Under  proposition  X  occurs  the  following  important  passage:] 
However  the  planets  have  been  formed  while  they  were  yet 
in  fluid  masses,  all  the  heavier  matter  subsided  to  the  centre. 

13 


MEMOIRS    ON 


Since,  therefore,  the  common  matter  of  our  earth  on  the  sur- 
face thereof  is  about  twice  as  heavy  as  water,  and  a  little  lower, 
in  mines,  is  found  about  three,  or  four,  or  even  five  times  more 
heavy,  it  is  probable  that  the  quantity  of  the  whole  matter  of 
the  earth  may  be  five  or  six  times  greater  than  if  it  consisted 
all  of  water.* 

[Under  propositions  XVIII.  and  XIX.,  Newton  proves  that 
the  axes  of  the  planets  are  hss  than  the  diameters  drawn  perpen- 
dicular to  the  axes.  He  shows  hoiv  centrifugal  force  acts  in 
determining  the  form  of  the  earth,  and  discusses  the  measurements 
of  terrestrial  arcs  known  at  that  time;  he  deduces  therefrom  that 
gravity  will  be  leb'ened  at  the  equator  by  ^-§  of  itself  ',  and  that 
the  earth  will  be  higher  at  the  equator  than  at  the  poles  by  17.1 
miles.] 

BOOK  III.     PROPOSITION  XX.     PROBLEM  IV. 

To  find  and  compare  together  the  weights  of  bodies  in  the  dif- 
ferent regions  of  our  earth. 

Because  the  weights  of  the  unequal  legs  of  the  canal  of  water 
are  equal  ;  and  the  weights  of  the  parts  proportional 
to  the  whole  legs,  and  alike  situated 
in  them,  are  one  to  another  as  the 
weights  of  the  wholes,  and  therefore 
equal  betwixt  themselves;  the  weights 
of  equal  parts,  and  alike  situated  in 
the  legs,  will  be  reciprocally  as  the 
legs  —  that  is,  reciprocally  as  230  to 
229.  And  the  case  is  the  same  in 
all  homogeneous  equal  bodies  alike 
situated  in  the  legs  of  the  canal. 
Their  weights  are  reciprocally  as  the 
legs  —  that  is,  reciprocally  as  the  dis- 
tances of  the  bodies  from  the  centre  of  the  earth.  Therefore, 
if  the  bodies  are  situated  in  the  uppermost  parts  of  the  canals, 
or  on  the  surface  of  the  earth,  their  weights  will  be  one  to  an- 
other reciprocally  as  their  distances  from  the  centre.  And,  by 
the  same  argument,  the  weights  in  all  other  places  round  the 
whole  surface  of  the  earth  are  reciprocally  as  the  distances  of 

*  [  This  was  a  wonderfully  good  guess  on  Newton's  part,  since  tJie  best  of  the 
later  determinations  give  about  5.5  for  the  mean  specific  gravity  oftJie  earth.} 

14 


THE    LAWS    OF    GRAVITATION 

the  places  from  the  centre;  and,  therefore,  in  the  hypothesis 
of  the  earth's  being  a  spheroid,  are  given  in  proportion. 

[Newton  then  states  that  ('the  lengths  of  pendulums  vibrating 
in  equal  times  are  as  the  forces  of  gravity";  he  enumerates  the 
experiments  on  the  periods  of  pendulums  made  at  different  parts 
of  the  earth's  surface,  and  tests  his  conclusions. 

The  folio  iving  remarks  appear  on  pp.  20-25  of  Motte's  transla- 
tion of  the  "  tie  Mundi  Systemate"  wherein  Newton,  after  a 
reference  to  his  pendulum  experiments,  given  on  p.  11  of  this 
volume,  says:} 

Since  the  action  of  the  centripetal  force  upon  the  bodies  at- 
tracted is,  at  equal  distances,  proportional  to  the  quantities  of 
matter  in  those  bodies,  reason  requires  that  it  should  be  also 
proportional  to  the  quantity  of  matter  in  the  body  attracting. 

For  all  action  is  mutual,  and  (by  the  third  law  of  motion) 
makes  the  bodies  mutually  to  approach  one  to  the  other,  and 
therefore  must  be  the  same  in  both  bodies.  It  is  true  that  we 
may  consider  one  body  as  attracting,  another  as  attracted;  but 
this  distinction  is  more  mathematical  than  natural.  The  at- 
traction is  really  common  of  either  to  other,  and  therefore  of 
the  same  kind  in  both. 

And  hence  it  is  that  the  attractive  force  is  found  in  both. 
The  sun  attracts  Jupiter  and  the  other  planets;  Jupiter  at- 
tracts its  satellites  ;  and,  for  the  same  reason,  the  satellites  act 
as  well  one  upon  another  as  upon  Jupiter,  and  all  the  planets 
mutually  one  upon  another. 

And  though  the  mutual  actions  of  two  planets  may  be  dis- 
tinguished and  considered  as  two,  by  which  each  attracts  the 
other,  yet,  as  those  actions  are  intermediate,  they  do  not  make 
but  one  operation  between  two  terms.  Two  bodies  may  be 
mutually  attracted  ea.ch  to  the  other  by  the  contraction  of  a 
cord  interposed.  There  is  a  double  cause  of  action,  to  wit,  the 
disposition  of  both  bodies,  as  well  as  a  double  action  in  so  far 
as  the  action  is  considered  as  upon  two  bodies  ;  but  as  betwixt 
two  bodies  it  is  but  one  single  one.  It  is  not  one  action  by 
which  the  sun  attracts  Jupiter,  and  another  by  which  Jupiter 
attracts  the  sun;  but  it  is  one  action  by  whicli  the  sun  and 
Jupiter  mutually  endeavour  to  approach  each  the  other.  By 
the  action  with  which  the  suii  attracts  Jupiter,  Jupiter  and 
the  sun  endeavour  to  come  nearer  together  (by  the  third  law  of 
motion);  and  by  the  action  with  which  Jupiter  attracts  the 

15 


MEMOIRS    ON 

i 

sun,  likewise  Jupiter  aud  the  sun  endeavour  to  come  nearer  to- 
gether. But  the  sun  is  not  attracted  towards  Jupiter  by  a 
twofold  action,  nor  Jupiter  by  a  twofold  action  towards  the 
sun ;  but  it  is  one  single  intermediate  action,  by  which  both 
approach  nearer  together. 

Thus  iron  draws  the  loadstone  as  well  as  the  loadstone 
draws  the  iron  ;  for  all  iron  in  the  neighbourhood  of  the  load- 
stone draws  other  iron.  But  the  action  betwixt  the  loadstone 
and  iron  is  single,  and  is  considered  as  single  by  the  philoso- 
phers. The  action  of  iron  upon  the  loadstone  is,  indeed,  the 
action  of  the  loadstone  betwixt  itself  and  the  iron,  by  which 
both  endeavour  to  come  nearer  together ;  and  so  it  manifestly 
appears,  for  if  you  remove  the  loadstone  the  whole  force  of  the 
iron  almost  ceases. 

In  this  sense  it  is  that  we  are  to  conceive  one  single  action 
to  be  exerted  betwixt  two  planets,  arising  from  the  conspiring 
natures  of  both;  and  this  action  standing  in  the  same  relation 
to  both,  if  it  is  proportional  to  the  quantity  of  matter  in  the 
one,  it  will  be  also  proportional  to  the  quantity  of  matter  in 
the  other. 

Perhaps  it  may  be  objected  that,  according  to  this  phil- 
osophy (prop.  74,  book  I),  all  bodies  should  mutually  attract 
one  another,  contrary  to  the  evidence  of  experiments  in  ter- 
restrial bodies ;  but  I  answer  that  the  experiments  in  terres- 
trial bodies  come  to  no  account ;  for  the  attraction  of  homo- 
geneous spheres  near  their  surfaces  are  (by  prop.  72,  book 
I)  as  their  diameters.  Whence  a  sphere  of  one  foot  in  diam- 
eter, and  of  a  like  nature  to  the  earth,  would  attract  a  small 
body  placed  near  its  surface  with  a  force  20,000,000  *  times  less 
than  the  earth  would  do  if  placed  near  its  surface  ;  but  so 
small  a  force  could  produce  no  sensible  effect.  If  two  such 
spheres  were  distant  but  by  one-quarter  of  an  inch,  they  would 
not,  even  in  spaces  void  of  resistance,  come  together  by  the 
force  of  their  mutual  attraction  in  less  than  a  month's  time;f 

*  [If  the  sphere  is  one  foot  in  diameter,  thin  number  should  be  40,000,000, 
since  the  diameter  of  the  earth  is  about  40, 000, 000  ft.  But  perhaps  Newton 
intended  to  say  a  sphere  of  one  foot  in  radius.  } 

f  [The  time  is  very  much  less.  On  the  assumption  that  each  of  the  spheres  is 
one  foot  in  diameter,  Poynting  (185,  p.  10)  finds  the  time  to  be  about  320  sec- 
onds. If,  however,  we  take  one  foot  as  the  radius  of  each  sphere,  Todhunter 
(140,  vol.  I,  p.  461)  *lunox  that  the  time  is  less  than  250  seconds.} 

16 


THE    LAWS    OF    GRAVITATION 


Fig.b 


and  less  spheres  will  come  together  at  a  rate  yet  slower,  viz., 
in  the  proportion  of  their  diameters.  Nay,  whole  mountains 
will  not  be  sufficient  to  produce  any  sensible  effect.  A  moun- 
tain of  an  hemispherical  figure,  three  miles  high  and  six  broad, 
will  not,  by  its  attraction,  draw  the  pendulum,  two  min- 
utes *  out  of  the  true  perpendicular;  and  it  is  only  in  the 
great  bodies  of  the  planets  that  these  forces  are  to  be  per- 
ceived, unless  we  may  reason  about  smaller  bodies  in  manner 
following.! 

Let  ABCD  represent  the  globe  of 
the  earth  cut  by  any  plane,  AC,  into 
two  parts,  ACB  and  ACD.  The  part 
ACB  bearing  upon  the  part  ACD 
presses  it  with  its  whole  weight ;  nor 
can  the  part  ACD  sustain  this  press- 
ure, and  continue  unmoved,  if  it  is 
not  opposed  by  an  equal  contrary 
pressure.  And  therefore  the  parts 
equally  press  each  other  by  their 
weights — that  is,  equally  attract  each 

other,  according  to  the  third  law  of  motion;  and,  if  separated 
and  let  go,  would  fall  towards  each  other  with  velocities  re- 
ciprocally as  the  bodies.  All  which  we  may  try  and  see  in  the 
loadstone,  whose  attracted  part  does  not  propel  the  part  at- 
tracting, but  is  only  stopped  and  sustained  thereby. 

Suppose  now  that  ACB  represents  some  small  body  on  the 
earth's  surface  ;  then,  because  the  mutual  attractions  of  this 
particle,  and  of  the  remaining  part  ACD  of  the  earth  towards 
each  other,  are  equal,  but  the  attraction  of  the  particle  towards 
the  earth  (or  its  weight)  is  as  the  matter  of  the  particle  (as  we 
have  proved  by  the  experiment  of  the  pendulums),  the  at- 
traction of  the  earth  towards  the  particle  will  likewise  be  as 
the  matter  of  the  particle;  and  therefore  the  attractive  forces  of 
all  terrestrial  bodies  will  be  as  their  several  quantities  of  matter. 

The  forces  (prop.  71,  book  I),  which  are  as  the  matter  in 

*  [Maskelyne  (31)  says  with  reference  to  this :  "It  will  appear,  by  a  very  easy 
calculation,  that  such  a  mountain  would  attract  the  plumb-line  1'  18"  from  the 
perpendicular."] 

\  [  This  paragraph  is  of  great  importance,  because  in  it  Newton  indicates 
the  methods  of  all  ilie  experiments  yet  made  in  order  to  measure  gravitational 
attraction  in  terrestrial  bodies.  ] 

B  17 


MEMOIRS    ON 

terrestrial  bodies  of  all  forms,  and  therefore  are  not  mutable 
with  the  forms,  must  be  found  in  all  sorts  of  bodies  whatsoever, 
celestial  as  well  as  terrestrial,  and  t>e  in  all  proportional  to  their 
quantities  of  matter,  because  among  all  there  is  no  difference 
of  substance,  but  of  modes  and  forms  only.  But  in  celestial 
bodies  the  same  thing  is  likewise  proved  thus.  We  have  shewn 
that  the  action  of  the  circumsolar  force  upon  all  the  planets 
(reduced  to  equal  distances)  is  as  the  matter  of  the  planets ; 
that  the  action  of  the  circumjovial  force  upon  the  satellites  of 
Jupiter  observes  the  same  law;  and  the  same  thing  is  to  be  said 
of  all  the  planets  towards  every  planet;  but  thence  it  follows 
(by  prop.  69,  book  I)  that  their  attractive  forces  are  as  their 
several  quantities  of  matter. 

As  the  parts  of  the  earth  mutually  attract  one  another,  so 
do  those  of  all  the  planets.  If  Jupiter  and  its  satellites  were 
brought  together,  and  formed  into  one  globe,  without  doubt 
they  would  continue  mutually  to  attract  one  another  as  before. 
And,  on  the  other  hand,  if  the  body  of  Jupiter  was  broken  into 
more  globes,  to  be  sure,  these  would  no  less  attract  one  another 
than  they  do  the  satellites  now.  From  these  attractions  it  is 
that  the  bodies  of  the  earth  and  all  the  planets  effect  a  spheri- 
cal figure,  and  their  parts  cohere,  and  are  not  dispersed  through 
the  aether.  But  we  have  before  proved  that  these  forces  arise 
from  the  universal  nature  of  matter  (prop.  72,  book  I),  and 
that,  therefore,  the  force  of  any  whole  globe  is  made  up  of  the 
several  forces  of  all  its  parts.  And  from  thence  it  follows  (by 
cor.  3,  prop.  74)  that  the  force  of  every  particle  decreases  in 
the  duplicate  proportion  of  the  distance  from  that  particle; 
and  (by  prop.  73  and  75,  book  I)  that  the  force  of  an  entire 
globe,  reckoning  from  the  surface  outwards,  decreases  in  the 
duplicate,  but,  reckoning  inwards,  in  the  simple  proportion  of 
the  distances  from  the  centres,  if  the  matter  of  the  globe  be 
uniform.  And  though  the  matter  of  the  globe,  reckoning  from 
the  centre  towards  the  surface,  is  not  uniform  (prop.  73,  book 
I),  yet  the  decrease  in  the  duplicate  proportion  of  the  distance 
outwards  would  (by  prop.  76,  book  I)  take  place,  provided  that 
difformity  is  similar  in  places  round  about  at  equal  distances 
from  the  centre.  And  two  such  globes  wil1  (by  the  same  prop- 
osition) attract  one  the  other  with  a  force  decreasing  in  the 
duplicate  proportion  of  the  distance  between  their  centres. 

Wherefore  the  absolute  force  of  every  globe  is  as  the  quan- 

18 


THE    LAWS    OF    GRAVITATION 

tity  of  matter  which  the  globe  contains ;  but  the  motive  force 
by  which  every  globe  is  attracted  towards  another,  and  which, 
in  terrestrial  bodies,  we  commonly  call  their  weight,  is  as  the 
content  under  the  quantities  of  matter  in  both  globes  applied 
to  the  square  of  the  distance  between  their  centres  (by  cor.  4, 
prop.  76,  book  I),  to  which  force  the  quantity  of  motion,  by 
which  each  globe  in  a  given  time  will  be  carried  towards  the 
other,  is  proportional.  And  the  accelerative  force,  by  which 
every  globe  according  to  its  quantity  of  matter  is  attracted 
towards  another,  is  as  the  quantity  of  matter  in  that  other  globe 
applied  to  the  square  of  the  distance  between  the  centres  of 
the  two  (by  cor.  2,  prop.  76,  book  I) ;  to  which  force  the  ve- 
locity by  which  the  attracted  globe  will,  in  a  given  time,  be 
carried  towards  the  other  is  proportional.  And  from  these 
principles  well  understood,  it  will  be  now  easy  to  determine 
the  motions  of  the  celestial  bodies  among  themselves. 


SIR  ISAAC  NEWTON  was  born  at  Woolsthorpe,  near  Grant- 
ham,  in  Lincolnshire,  in  1642.  He  was  educated  at  the  Grant- 
ham  grammar-school,  entered  Trinity  College,  Cambridge,  in 
1661,  and  received  his  degree  four  years  later.  He  at  once 
began  to  make  those  magnificent  discoveries  in  mathematics 
and  physics  which  have  made  his  name  immortal.  In  1665  he 
committed  to  writing  his  first  discovery  on  fluxions,  and  shortly 
afterward  made  the  unsuccessful  attempt,  to  which  we  have 
already  referred,  to  explain  lunar  and  planetary  motions.  He 
next  turned  his  attention  to  the  subject  of  optics  ;  his  work  in 
that  field  includes  the  discovery  of  the  unequal  refrangibility 
of  differently  coloured  lights,  the  compositeness  of  white  light 
and  chromatic  aberration.  Having  erroneously  concluded  that 
this  aberration  could  not  be  rectified  by  a  combination  of  lenses, 
he  turned  his  attention  to  reflectors  for  telescopes  and  made  a 
great  advance  in  that  direction.  His  name  is  also  closely  iden- 
tified with  the  colours  due  to  thin  plates.  From  1669  to  1701 
he  was  Lucasian  professor  of  mathematics  at  Cambridge.  He 
was  elected  to  membership  in  the  Royal  Society  in  1671,  and 
from  1703  until  his  death  was  its  president ;  he  became  a  mem- 
ber of  the  Paris  Academy  in  1699.  The  publication  of  his  work 
on  Optics  had  caused  some  controversy,  and  such  a  lover  of 
peace  was  Newton,  and  so  little  did  he  care  for  the  praise  of 

19 


MEMOIRS  ON  THE  LAWS  OF  GRAVITATION 

the  world,  that  it  was  only  at  the  earnest  solicitation  of  Hal  ley 
that  he  was  willing  to  give  to  the  public  the  results  of  his  won- 
derful researches  on  central  orbits,  and  universal  gravitation  ; 
these  included  an  explanation  of  the  lunar  inequalities,  the 
figure  of  the  earth,  the  precession  of  the  equinoxes  and  the 
tides,  and  a  method  of  comparing  the  masses  of  the  heavenly 
bodies.  In  1669  he  became  a  member  of  Parliament,  in  1696 
Warden  of  the  Mint,  and  from  1699  until  his  death  was  Master 
of  the  Mint.  He  gave  much  valuable  aid  in  the  recoinage  of 
the  money  and  in  questions  of  finance  at  this  period.  He  was 
knighted  in  1705.  During  the  latter  years  of  his  life  much  of 
his  time  was  devoted  to  his  public  duties.  He  died  in  1727, 
and  was  buried  in  Westminster  Abbey. 

20 


LA  FIGURE  DE  LA  TERRE 

Determines  par  les  Observations  cle  Messieurs  Bouguer,  et 
de  la  Condamine,  de  1'Academie  Royale  des  Sciences,  envoyes 
par  ordre  du  Roy  au  Perou,  pour  observer  aux  environs  de 
1'fCquateur. 

Avec  une  Relation  abregee  de  ce  Voyage,  qui    contient  la 
description  du  Pays  dans  lequel  les  operations  ont  ete  faites. 

PAR  M.  BOUGUER 
A    Paris,  1749 


Section  7,   pp.  327-394 


THE  FIGURE   OF    THE  EARTH 

Determined  by  the  observations  of  MM.  Bouguer  and  de  la 
Condamine,  of  the  Royal  Academy  of  Sciences,  sent  to  Peru 
by  order  of  the  King  to  make  observations  near  the  equator. 

With  a  brief  account  of  their  travels  and  a  description  of  the 
country  in  which  the  investigations  were  made. 

BY   PIERRE    BOUGUER 
Paris,   1749 

Section  7    pp.  327-394 


21 


CONTENTS  OF  SECTION  VII 

PAGE 

Introduction 23 

Chap.  I. — Experiments  Made  in  Order  to  find  the  Length  of  the  Seconds- 
Pendulum  24 

Description  of  Pendulum 24 

Method  of  Observation  (Omitted). 

Observed  Lengths  of  Seconds- Pendulum  at  Various  Places 25 

Corrections  to  be  Made  in  the  Observed  Lengths 25 

Corrected  Lengths  of  Seconds-Pendulum  at  Various  Places 27 

Chap.  II. — Comparison  of  Attraction  and  Centrifugal  Force  (Omitted). 
Chap.  III. — Remarks  on  the  Diminution   in  Attraction  at  Different 

Heights  above  Sea-level 27 

Calculation  of  the  Attraction  Due  to  a  Plateau 29 

Deduction  of  the  Mean  Density  of  the  Earth  from  Pend- 
ulum Experiments. 32 

Chap.  IV. — On  the  Deflection  of  the  Plumb-line  by  a  Mountain ...  33 

Description  of  Mount  Chimborazo 34 

Its  Deflection  of  the  Plumb-line  Calculated  from  the  Theory. .  34 

Various  Ways  Suggested  for  Showing  the  Deflection 35 

Description  of  the  Method  Employed 38 

Examination  of  the  Attraction  of  Chimborazo 39 

Meridian  Altitudes  at  the  First  Station 40 

Measurements  Made  to  find  the  Relative  Positions  of  the  two 

Stations 41 

Meridian  Altitudes  at  the  Second  Station  (Omitted). 

Corrected  Meridian  Altitudes  at  the  Second  Station 42 

Calculations  for  the  Observed  Deflection  of  the  Plumb-line. . .  42 

Its  Poor  Agreement  with  that  Calculated  from  Theory 43 

Appendix  (Omitted). 


SECTION    VII.    OP    BOUGUER'S    FIGURE 
OF    THE    EARTH 

ACCOUNT    OF    THE    EXPERIMENTS    OR   OBSERVATIONS    ON   GRAV- 
ITATION,   WITH    REMARKS    ON   THE   CAUSES    OF 
THE   FIGURE   OF   THE    EARTH 

1.  HAVING  discussed  everything  that  bears  on  the  earth  con- 
sidered as  a  geometrical  body,  it  remains  for  us,  before  terminat- 
ing this  work,  to   verify  the  facts  which  give   us  some  slight 
knowledge  of  the  interior  conformation  of  this  great  mass  con- 
sidered as  a  physical  body.  .  .  . 

2.  The  first  question  which  presents  itself  on  this  matter  is 
a  consideration  of  the  part  played  in  the  flattening  of  the  earth 
by  the  attraction  which  compresses  it  from  all  sides,  urging  all 
masses  towards  certain  points.     We  know,  since  M.  Eicher  first 
remarked  it  (in  1672  in  Cayenne),  that  this  force  is  not  every- 
where the  same.     It  is  greater  towards  the  poles,  and  less  to- 
wards the  equator.    This  agrees  perfectly  with  the  figure  of  the 
earth,  which  appears  to  have  yielded  a  little  to  the  great  press- 
ure at  the  poles,  and  to  be  slightly  elevated,  on  the  contrary, 
at  the  equator,  where  the  compressing  force  was  more  feeble. 
But  does  the  effect  correspond  exactly  to  the  cause  upon  which 
we  desire  it  to  depend?     Is  the  difference  in  attraction  so  great 
that  we  can  attribute  to  it  all  the  inequality  which  exists,  as 
we  have  seen,  between  the  two  diameters  of  our  globe  ?     To 
answer  this  question  it  is  necessary  to  determine,  by  exact  ex- 
periment, how  much  the  attraction  actually  differs  in  different 
parts  of  the  earth.  .  .  .  We  have  two  methods  for  observing 
the  change  in  attraction  as  we  pass  from  one  region  to  another  ; 
we  have  only  to  examine  how  much   more  quickly  or  more 
slowly  a  pendulum  of  given  length  oscillates ;  or  else  to  find 
the  length  of  the  pendulum  whose  time  of  vibration  is  exactly 

23 


MEMOIRS    ON 

a  second  ;  the  differences  which  we  shall  find  in  the  length  of 
this  pendulum  will  determine  the  changes  of  the  attraction  as 
we  go  from  one  region  to  another. 


ACCOUNT    OF    THE    EXPERIMENTS    MADE    FOR    THE    PURPOSE    OF 
DETERMINING    THE    LENGTH    OF    THE    SECONDS-PENDULUM 

3.  My  first  experiments  with  the  pendulum  were  made  at- 
Petit-Goave  in  the  island  of  St.  Domingue.     They  are  reported 
in  the  memoirs  of  the  Academy  for  1735  and  1736.  .  .  . 

4.  The  instrument  which  I  almost  always  used,  and  which  I 
still  use,  is  extremely  simple.     I  make  the  pendulum  always 
exactly  of  the  same  length,  and  I  compare  its  oscillations  with 
those  of  a  clock  which  I  regulate  by  daily  observations.     It  is 
not,  properly  speaking,  by  the  different  lengths  of  the  pend- 
ulum that  I  judge  of  the  intensity  of  gravitation  at  different 
places  ;  I  judge  of  it  only  by  the  greater  or  less  rapidity  of  the 
oscillations,  or  by  the  number  of  oscillations  made  by  the  pend- 
ulum in  24  hours.  ...  It  appears  to  me  to  be  much  easier  to 
count  the  number  of  oscillations  than  to  measure  directly  dif- 
ferences of  a  few  hundredths  of  a  line*  in  the  length  of  the 
pendulum. 

[Then  follows  an  account  of  his  pendulum.  The  bob  was  of 
copper,  composed  of  tzvo  equal  truncated  cones  joined  at  their 
greater  bases.  The  thread  ivas  a  fibre  of  aloe,  which  is  not  af- 
fected by  the  weather.  The  length  was  maintained  constant  by 
having  it  always  so  that  an  iron  rule  just  fitted  in  between  the 
clamp  and  the  bob.  The  length  of  the  equivalent  simple  pendulum 
was  36  pouces,  7.015  lines. 

Bouguer  gives  a  description  of  a  scale  fixed  behind  the  pend- 
ulum,  by  means  of  which  he  could  observe  the  decrement  and  the 
time  required  by  the  pendulum  to  gain  an  oscillation  on  the 
clock.  ] 

10.  It  is  time  to  relate  the  experiments.  ...  I  shall  choose 
one  of  those  which  I  made  on  the  rocky  summit  of  Pichincha 
[2434  toises  above  sea-level],  in  the  month  of  August,  1737.  The 

*  [72  ponces  —  \  toise  =  1.949  metres  =  6.3945  ft,     12  lines  =  1  pouce  ] 

24 


THE    LAWS    OF    GRAVITATION 

force  of  attraction  Was  feeble,  not  only  because  we  were  nearly 
over  the  equator  at  this  place,  but  also  because  we  were  at  a 
very  great  height  above  the  surface  of  the  sea.  .  .  . 
[Details  of  experiment.] 

12.  .  .  .  We  find  in  this  way  that  the  pendulum  which  beats 
seconds  at  the  equator,  and  in  the  highest  accessible  place  on 
the  earth,  is  36  polices  6.69  lines  in  length.     I  made  other  ex- 
periments at  the  same  place  which  agreed  as  exactly  as  possible 
with  this  result.     [One  made  by  Don  Antonio  de  Ulloa  gave  36 
pouces,  6.715  lines.      We  may  take  as  the  mean  36  pouces,^.7Q 
lines.  ] 

13.  I  have  found  by  the  same  proceedings  and  with  the  aid 
of  the  same  instruments,  the  length  of  the  seconds-pendulum 
at  Quito  [1466  toises  above  sea-level],  to  be  36  ponces,  6.82  or 
6.83  lines.     I  have  verified  it  at  different  times  and  in  all  sea- 
sons of  the  year:  at  times  of  aphelion  and  perihelion,  at  the 
equinoxes,  and  when  the  sun  was  at  intermediate  points;  the 
extreme  results  were  36  pouces,  6.79  lines  and  6.85  lines,  with 
no  differences  which  could  not  be  attributed  to  the  inevitable 
errors  of  observation.  .  .  . 

[  The  question  of  a  possible  yearly  change  is  discussed. 

Experiments  were  made  with  the  same  apparatus,  in  1740,^ 
I' Isle  de  I'Inca,  14'  or  15'  from  the  equator,  and  scarcely  40  toises 
above  sea-level.     Bouguer  regards  this  determination  as  that  of 
the  true  equinoctial  pendulum.] 
15. 


Place 

Length  found  by  experiment 

i  2434  toises  absolute  height. 
Under  the  equator  at  •!  1466     "             " 
/  Sea-level 

36  pouce»,  6.70  lines. 
6.83      " 
707      " 

At  Portobello   9°  34'  N   latitude 

7  16      " 

At  Petit-Goave,  18°  27'  "         "     
At  Paris 

7.33      " 

8.58      •' 

CORRECTIONS    WHICH     MUST    BE    APPLIED    TO    THE    LENGTH    OF 

THE    PENDULUM    AS    DETERMINED    DIRECTLY    FROM 

THE    EXPERIMENTS. 

16.   [Bouguer  remarks  that  these  corrections  arise  from  changes 
in  tem/oerature  and  in  the  constitution  of  the  atmosphere.']     The 

35 


MEMOIRS    ON 

first  cause  does  riot  really  change  the  length,  it  only  makes  it 
appear  different  according  as  the  measures  we  use  are  differ- 
ently altered  by  heat  or  cold;  but  the  other  cause  brings  in  a 
real  inequality,,  since  it  produces  nearly  the  same  effect  as  if 
the  weight  were  greater  or  smaller.  .  .  . 

17.  ...  Since  the  temperature  of  Quito  does  not  differ  from 
that  of  Paris  in  the  middle  of  spring,  we  have  only  to  refer  all 
our  results  to  it.     That  is,  without  altering  the  lengths  of  the 
pendulum  found  in  these  two  cities,  we  have  only  to  correct  all 
the  others  by  increasing  or  diminishing  them,  according  as  the 
metal  rules  we  used  were  expanded  by  the  heat  or  contracted 
by  the  cold.     [He  concludes  from  his  experiments  that  a  change 
of  length  of  pendulum  of  .02  lines  corresponds  to  a  change  of 
temperature  of  3°R.    Hence  he  had  to  add  .075  lines  to  the  length 
found  at  sea-level,  and  subtract  .05  lines  from  that  found  at 
Pichincha.] 

18.  There  is  little  more  difficulty  in  finding  the  alteration  in 
the  length  of  the  pendulum  caused  by  the  medium  in  which 
the  experiments  are- made.   This  medium, whether  rare  or  dense, 
has  a  certain  weight,  and  that  of  the  small  mass  of  copper,  of 
which  the  bob  of  the  pendulum  is  formed,  is  a  little  lessened 
by  it.    The  small  mass  tends  to  fall  to  the  earth  with  only  the 
excess  of  its  weight  above  that  ot  the  air  which  surrounds  it. 
Thus  our  pendulums  are  acted  on  by  a  force  a  little  less  than  if 
we  had  performed  the  experiments  in  vacuo :  and  the  length 
of  the  seconds-pendulum,  which  we  found  directly  from  experi- 
ment, is  a  little  too  short  in  the  same  proportion. 

19.  The  use  of  the  barometer  enables  us  to  find  the  ratio  be- 
tween the  weight  of  mercury  and  of  air  in  all  the  parts  of  the 
atmosphere  which  are  accessible.     We  observe  how  many  feet 
it  is  necessary  to  ascend  or  descend  in  order  to  change  the 
height  of  the  mercury  by  a  line.  ...   I  have  found  in  this  way 
that  it  was  only  necessary  to  express  the  first  (the  weight  of 
air)  by  unity,  at  the  summit  of  Pichincha,  if  one  expressed  that 
of  copper  by  11000.   ...   So  I  always  found  the  seconds-pend- 
ulum too  small  by  y^^th  part.      To  correct    for  this    error 
we  must  add  .04  lines  [at  Pichincha  ;  .05  at  Quito  ;  .06  at  sea- 
level.]     .  .  .  This  is  the  first  time  that  any  one  has  taken  ac- 
count of  this  small  correction  which  enters  into  the  experi- 
ments, but  we  cannot  neglect  it  if  we  wish  to  attain  the  greatest 
accuracy.  .  .  . 

26 


THE    LAWS    OF    GRAVITATION 

[Bouguer  then  proves  that  the  time  of  vibration  is  not  appreci- 
ably affected  by  the  resistance  of  the  air.]* 

22.  Corrected  lengths  of  the  seconds-pendulum,  or  such  as 
they  would  be  if  the  oscillations  were  made  in  vacuo. 


Place. 


Under  the  equator  at  •]  1466 

(  Sea-level  . 
At  Portobello,  9°  34'  N.  latitude 
At  Petit-Goave,  18°  27'  "  "  . 
At  Paris  . 


2434  toises  absolute  height. 


pouces,6.69  lines. 
"       6.88     " 
"       7.21     " 
"       7.30     " 

7.47    " 
"       8.67     " 


IT 

COMPARISON     OF    ATTRACTION    AND    THE    CENTRIFUGAL    FORCE 

WHICH    BODIES    ACQUIRE   BY  THE    MOTION   OF  THE   EARTH 

ABOUT    ITS    AXIS,   WITH    REMARKS    ON    THE    EFFECTS 

OF   THESE    TWO    FORCES. 

[Bouguer  finds  that  the  primitive  attraction  (that  is,  the  attrac- 
tion the  earth  would  have  if  it  were  at  rest)  is  to  the  centrifugal 
force  as  288-JJ  :  1.  He  gives  a  table  showing  the  decrease  in  the 
length  of  the  seconds-pendulum  at  various  latitudes,  due  to  the 
centrifugal  force.  The  following  headings  will  give  an  idea  of 
the  matter  contained  in  the  rest  of  this  chapter.] 

The  centrifugal  force  produced  by  the  motion  of  the  earth 
about  its  axis  is  not  sufficient  to  produce  the  observed  differ- 
ences in  weight. 

The  primitive  attraction  does  not  tend  towards  a  common 
point  as  centre. 


Ill 

REMARKS    ON    THE    DIMINUTION    IN    THE    ATTRACTION    AT    DIF- 
FERENT   HEIGHTS    ABOVE    THE    LEVEL    OF    THE    SEA. 

40.  The   experiments  with   the    pendulum   which   we   have 
made  at  Quito  and  on  the  summit  of  Pichincha  teach  us  that 

*  [See  note  on  page    66] 

27 


MEMOIRS    ON 

the  attraction  changes  with  the  distance  from  the  centre  of  the 
earth.  This  force  goes  on  diminishing  as  we  ascend;  I  have 
found  the  pendulum  at  Quito  to  be  shorter  than  at  sea-level  by 
.33  lines,  or  the  y-gVr^1  P111^ :  Jin^  in  mounting  to  the  summit  of 
Pichincha  the  pendulum  is  shortened  again  by  .19  lines,  and  is 
8TTtn  Part  shorter  than  at  sea-level.*  One  cannot  attribute 
these  differences  to  the  centrifugal  force,  which,  being  greater 
the  higher  we  ascend,  ought  to  diminish  a  little  further  the 
primitive  attraction.  The  centrifugal  force  is  increased  by  the 
height  of  the  mountain  by  the  -j-^j-yth  part  only,  and  as  it  is 
itself  but  the  ^-J^th  part  of  the  weight,  it  is  clear  that  its  new 
increase  corresponds  to  .001  lines  only  in  the  length  of  the 
pendulum,  and  so  does  not  sensibly  contribute  to  the  dimi- 
nution of  the  other  force. 

41.  If  we  compare  the  shortening  which  the  pendulum  re- 
ceives with  the  height  at  which  the  experiment  was  made,  we 
see  that  the  forces  do  not  decrease  in  the  simple  inverse  ratio 
of  the  distances  from  the  centre  of  the  earth,  but  that  they 
follow  rather  the  proportion   of    the   square.     Quito  is  146(5 
toises  above  sea-level,  or  7^7 th  of  the  radius  of  the  earth  ;  but 
it  has  been  found  that  the  attraction  is  less  by  a  fraction  much 
more  considerable — namely,  by  a  TTsr^h  part,  which  is  nearly 
double  ;    this  is  not  very  far    from  the  inverse  ratio  of    the 
square  of  the  distance.   .   .   .   We  have  a  second  example  in  the 
experiment  made  on  Pichincha.     The  absolute  height  of  this 
mountain,  which  is  2434  toises  above  sea-level,  is  j^Vs^1  of  the 
radius  of  the  earth.    The  diminution  of  the  length  of  the  pend- 
ulum, or  of  the  attraction,  ought  then  to  be  the  grjth  part, 
if  it  is  to  be  in  the  inverse  ratio  of  the  square  of  the  distance  ; 
but  it  was  by  no  means  so  great — in  fact,  only  the  g-J-^th  part. 

42.  This  diminution  in  attraction,  as  we  go  above  sea-level,  is 
quite  in  conformity  with  what  we  otherwise  know.     We  can 
compare   with    the   attraction   here   experimented    upon   that 
which  keeps  the  moon  in  its  orbit,  or  which  obliges  it  con- 
tinually to  perform  a  circle  about  us.     These  two  forces  aro 
exactly  in  the  inverse  ratio  of  the  squares   of  the  distances 
from  the  centre  of    the  earth.     We  can   make  the  same  ex- 

*  [Pendulum  observations  were  made  at  these  and  other  places  in  Peru  by  dr 
la  Condamine  also  (8,  pp.  70,  144,  162-169).  For  a  complete  biblior/raphy  of 
pendulum  experiments,  see  that  published  by  La  Societe  Fran$aise  de  Physique 
(178,  vol.  4).] 

28 


THE    LAWS    OF    GRAVITATION 


amination  with  respect  to  the  principal  planets  which  have 
several  satellites,  or  with  respect  to"  the  sun,  towards  which 
all  the  principal  planets  are  attracted,  and  we  shall  always 
find  the  law  of  the  square.  Why,  then,  do  our  experiments 
constantly  give  a  law  not  entirely  in  agreement  with  this? 
Is  it  necessary  to  attribute  the  difference  to  some  error  on 
our  part ;  or  can  it  be  that  in  the  neighborhood  of  great  masses 
like  the  earth  the  law  under  consideration  is  observed  in  an 
imperfect  manner  only  ? 

43.  We  shall  find  ourselves  in  a  position  to  solve  this  diffi- 
culty, perhaps,  by  remarking  that  the  Cordilleras,  on  which  we 
were  placed,  form  a  kind  of  plateau,  or,  what  in  certain  ways 
amounts  to  the  same  thing,  the  surface  of  the  earth  is  there 
carried  to  a  greater  height  or  to  a  greater  distance  from  the 
centre.      There  is   reason    for   believing   that  in  this  second 
case  the  attraction  would  be  a  little  greater  ;  for  it  is  natural 
to  think  that  it  depends  upon  the  size  of  the  attracting  mass. 
There  are  then  two  things   to  be  considered   in  the  case  of 
the   experiments   on   the   pendulum  which   I   have  reported. 
These  experiments  were  made 

at  a  great  height  above  the  av- 
erage surface  of  the  earth,  and 
therefore  the  attraction  ought 
to  be  found  a  little  less.  But, 
on  the  other  hand,  the  group 
of  mountains  on  which  Quito  is 
placed  and  on  which  Pichincha 
rises,  and  all  the  other  sum- 
mits to  which  it  acts  as  a 
plinth,  ought  to  produce  nearly 
the  same  effect  as  if  the  earth 
at  this  place  were  larger  or  had 
a  greater  radius.  The  attrac- 
tion on  this  account  ought  to  increase.  Thus  it  depends  on 
a  kind  of  chance,  or,  to  speak  more  philosophically,  it  de- 
pends on  circumstances  which  we  do  not  yet  know,  whether 
the  attraction  at  Quito  will  be  equal  to  that  at  sea-level,  or  be 
smaller  or  larger. 

44.  Suppose  that  the  circle   ADD  represents   the  circum- 
ference of  the  earth,  of  which  C  is  the  centre,  and  that  Aa 
is   the   amount   by  which   Quito,,  situated    at    a,   is    elevated 

29 


Fig.  c. 


MEMOIRS    ON 

above  sea -level.  Imagine  a  new  spherical  shell  of  terrestrial 
matter,  occupying  all  the  interval  between  the  two  concen- 
tric surfaces  ADD  and  add;  or,  which  comes  to  the  same 
thing,  imagine  that  the  earth  increases  in  radius,  and  that 
Quito,  without  changing  its  position,  remains  at  the  level 
of  the  sea,  now  supposed  much  higher.  There  is  every  reason 
to  think  that  the  attraction  at  Quito  would,  as  a  consequence, 
be  found  greater  than  it  actually  is  at  A  or  at  D,  in  the 
ratio  of  CA  to  Ca.  It  is  necessary  for  that,  however,  to  sup- 
pose that  the  layer  of  earth  enclosed  between  the  two  con- 
centric surfaces  is  of  the  same  density  as  all  the  rest ;  for  if 
the  density  were  different  the  increase  would  no  longer  be  in 
the  same  ratio. 

45.  Call  r  the  radius,  and  A  the  density  of  the  earth.    Then 
rA  is  the  attraction   at  all   the  points  A,   D,  etc.,  supposing 
that  the  earth  ends  there.      Call  h  the  height  Aa,  which  is 
very  small  compared   with  r.      Then    the   attraction  at  a  is 
less  than  at  A,  in  the  ratio  of  r2  :  (r-f  7*)2,  or  its  diminution 
will  be  as  2h:r;   that  is,  if   the  attraction  is  rA  at  A,  it  is 
(r  — 2A)A  at  a,  and  this  supposes  that  the  earth  has  CA  only 
for  effective  radius.      But  all  this  will  be  subject  to  change 
if  we  add  to  our  globe  the  layer  Ac?D,  whose  density  is    5. 
This  new  spherical  layer,  if  it  had  the  same  density  as  the 
rest,  would  augment  the  attraction  at  the  surface  in  the  same 
ratio  as  the  radius  of  the  earth  became  greater.    The  increase 
would  be  in  the  ratio  of  r  :  r  +  U. 

46.  Thus  the  added  layer  would  not  only  make  up  for  the 
decrease  which   the   attraction   actually  suffers  when  we   go 
away  from  the  earth,  in  rising  by  the  height  Ka  —  h,  but  would 
add  a  new  amount  to  it,  equal  to  half  the  diminution,  since 
it  would  make  this  attraction,  which  is  actually  r  —  2h  at  the 
point  «,  become  r  +  Ji.     It  follows  that  the  attraction  which 
the  spherical  layer  can  produce  at  its   exterior   surface  at  a 
is  expressed  by  3A,  or  three  times  its  thickness;  but  we  must 
multiply  by  the  density  S,  because  we  suppose  that  the  den- 
sity of   the  layer  and  that   of  the  earth  as  a  whole  are   not 
equal. 

47.  To  recapitulate:  When  the  earth  has  its  radins,  CA=:r, 
the  attraction  at  A  is  rA,  and  at  the  height  h  is  (r  —  2&)A. 
But  when  we  add  to  the  earth  the  spherical  layer  Ae?D,  the 
attraction  at  a  becomes  (r  —  2 

30 


THE    LAWS    OF    GRAVITATION 

48.  All  that  remains  now  to  be  remarked  is  that  the  Cordill- 
eras of  Peru,  however  great  they  may  be,  ought  not  to  produce 
the  same  effect  as  the  spherical  shell  which  we  have  assumed. 
If  the  base  EE  of   the   Cordilleras    were    exactly  double   its 
height,  and  this  mass  had  the  shape  of  the  roof  of  a  house  of 
indefinite  length,  then  the  Cordilleras  would  produce  at  a  only 
\  the  effect  of  the  entire  spherical  shell,  as  can  be  easily  proved. 
But  there  are  further  additions  to  be  made  in  order  to  give  a 
more  accurate  idea  of  the  Cordilleras  of  Peru.    The  base  EE  is 
80  or  100  times  greater  than  the  height  Aa,  which  augments 
the  effect  in  precisely  the  same  ratio  as  the  angle   at  a   is 
greater.     This  angle  is  only  90°  when  we  find  the  effect  ^  of 
that  which  the  whole  spherical  layer  would  produce,  but  on 
account  of  the  great  width  of  the  base  of  the  Cordilleras  the 
angle  is  nearer  170°,  which  doubles  the  effect.     Moreover,  the 
Cordilleras  do  not  terminate  at  the  height  of  Quito  in  a  single 
summit  like  the  ridge  of  a  house  ;  it  is,  on  the  contrary,  quite 
10  or  12  leagues  broad  there.     One  can  suppose  then,  without 
fear  of  mistake,  that  the  effect  is  the  greatest  which  can  be 
produced  by  a  chain  of  mountains.     It  is  the  J  of  that  which  a 
spherical  layer  would  produce,  or  f//3,  and  if  we  add  to  it  the 
attraction  (r  —  2/z)A,  which   the  globe  ADD  produces  at  a,  we 
shall  have  ( r— 2A)A  -f  f/^*  as  the  expression  for  the  attraction 
at  Quito,  when  rA  expresses  that  at  sea-level. 

49.  The  difference  between   the   two    is  2AA  —  fAS,  which 
furnishes  the  subject  of  divers  quite  curious  remarks.     If  the 
matter  of  the  Cordilleras  were  more  compact  than  that  of  the 
average  of  the  whole  earth,  and  their  densities  were  as  4  :  3, 
the  difference  27iA  —  |7i<J  would  become  zero,  and  the  attrac- 
tion at  Quito  would  be  the  same  as  at  sea-level.     If  the  density 
B  were  still  greater,  our  expression  for  the  diminution  would 
change  sign  and  become  an   increase,  so  that  the  pendulum 
would  be  longer  at  Quito  than  at  sea-level.     But  it  is  evident 

*  [This  formula  is  independently  foundby  D' Alembert  (\Z,  vol.  6,  pj9. 85-92), 
by  Young  (51  and  95,  vol.  2,  p.  27),  and  by  Poixson  (65,  vol.  1,  pp.  492-6). 

Under  the  form  g*=  g0  (l  -  |7i  +    |^  it  is  known  as  "Dr.  Young'- 

Rule"  where  g}  is  the  value  of  gravity  at  height  h.  and g0  is  the  value  at  the 
sea-level.  Faye  (147)  contends  that  the  last  term  of  the  equation  sliould  be  left 
out ;  and  if  Airy' 8  "flotation  theory"  (94),  or  Faye's  compensation  theory 
(146-1),  be  true,  there  is  no  doubt  thai  this  term  requires  correction.] 

31 


MEMOIRS    ON 

that  things  are  riot  so.  The  difference  in  the  length  of  the  pend- 
ulum is  sufficiently  great  to  let  us  see  that  the  density  of  the 
matter  of  which  the  Cordilleras  is  formed  is  much  smaller  than 
that  of  the  rest  of  the  globe. 

50.  We  have  found  by  experiment  a  diminution  of  a  -j-g^th 
part  in  the  length  of  the  pendulum,  or  in  the  attraction,  as  we 
go  from  the  sea-level  to  Quito.  So  T^T  corresponds  to  27*A  — 
%M,  as  compared  with  rA,  which  expresses  the  attraction  at 


sea-level  ;  that  is,  we  have  —  L-  =  a/*A  ~  %M.     If  we  put  -  = 

Idol  rA  r 

which  is  the  ratio  of  the  height  of  Quito  to  the  radius 


of  the  earth,  we  shall  have  —  —  =  —  -J—  x  ?^—  1?«    Whence 

lool        <iZoi  A 


850 
we  deduce  3  =  —  —  A,  which  tells  us  that  the  Cordilleras   of 

OiJtJ'J 

Peru,  in  spite  of  all  the  minerals  they  contain,  have  less  than  J 
the  density  of  the  interior  of  the  earth.* 

51.  We  admit  that  this  determination  may  contain  a  few  er- 
rors on  account  of  the  large  number  of  elements  we  had  to  em- 
ploy in  order  to  arrive  at  it.  Nevertheless,  if  we  once  admit 
that  the  attraction,  when  the  other  circumstances  are  the  same, 
follows  exactly  the  direct  ratio  of  the  masses,  we  cannot  doubt 
that  the  Cordilleras  of  Peru  have  a  density  considerably  less 
than  that  of  the  rest  of  the  globe.  If  we  suppose  A  and  3  equal, 
our  expression  for  the  difference  of  the  attractions  at  Quito  and 

at  sea-level  would  become  -^-  A  ;  which  would  make  the  differ- 

2r 

ence  between  the  lengths  of  the  pendulum  4  times  too  small,  or 
the  attractions  as  the  square  roots,  instead  of  the  squares,  'of 
the  distances  from  the  centre  of  the  earth.  The  attraction  at 
Quito  would  be  less  than  at  sea-level  by  only  the  •j-^Tj'th  part, 
and  the  pendulum  would  be  really  shorter  by  only  9  or  10  hun- 
dred ths  of  a  line,  and  in  appearance  by  2  or  3,  on  account  of  the 
different  constitution  and  temperature  of  the  air.  The  differ- 
ence of  the  lengths  of  the  pendulum  is  certainly  greater.  Thus 
it  is  necessary  to  admit  that  the  earth  is  much  more  compact 

*  [To  give  Bouguer's  result  more  accurately,  the  density  of  the  earth  is  4.7 
times  that  of  the  Cordilleras.     Saigey  (74,  p.  149)  has  made  a  recalculation  of 
these  results,  with  the  proper  reduction  to  nacuo,  and  finds  4.25.     He  has  done 
the  same  for  de  la  Condamine's  pendultim  experiments,  with  a  result  4.50. 
For  Addendum,  see  p.  160.] 


THE    LAWS    OF    GRAVITATION 

below  than  above,  and  in  the  interior  than  at  the  surface.  For 
the  soil  of  Quito  is  like  that  of  all  other  countries  ;  it  is  a  mixt- 
ure of  earth  and  stones,  with  some  metallic  constituents.  .  .  . 
Those  physicists  who  imagined  a  great  void  in  the  middle  of 
the  earth,  and  who  would  have  us  walk  on  a  kind  of  very  thin 
crust,  can  think  so  no  longer.  We  can  make  nearly  the  same 
objection  toWoodward's  theory  of  great  masses  of  water  in  the 
interior.  Bat  let  us  continue  to  limit  ourselves  to  the  facts,  or 
to  the  only  immediate  deductions  which  we  can  draw  from  them. 
These  deductions  are  confirmed  by  the  observations  described 
in  the  next  chapter,  which  is  in  the  form  I  gave  it  in  Peru  be- 
fore forwarding  it  to  France. 


IV 

MEMOIR    ON    ATTRACTION    AND    ON    THE    MANNER   OF   OBSERV- 
ING   WHETHER    MOUNTAINS    EXHIBIT    IT    (READ    AT   THE 
ACADEMIE    DES    SCIENCES,   IN    OCTOBER,  1739) 

52.  It  is  very  difficult  not  to  accept  attraction  as  a  principle 
of  fact  or  of  experience.  The  most  rigid  Cartesians,  like  all 
other  philosophers,  cannot  dispense  with  it  in  this  sense.  All 
they  can  do  is  to  reserve  to  themselves  the  right  of  explaining 
it.  ...  Since  all  the  planets  circle  about  the  sun,  there  must 
necessarily  be  a  force,  I  shall  not  say  shoving  them  or  drawing 
them,  but  rather  transporting  them  at  each  moment  towards 
this  star.  .  .  .  Nothing  prevents  us  from  giving  to  this  force 
the  name  "  attraction,"  and  from  trying  to  assign  to  it  a  phys- 
ical cause.  .  .  . 

[Bouguer  affirms  that  in  establishing  a  new  principle,  it  is  not 
only  necessary  to  prove  the  insufficiency  of  all  others,  but  their 
impossibility  also.  ] 

54.  While  waiting  for  all  this  to  happen,  it  will  contribute 
to  the  perfection  of  physics  if  we  examine  more  carefully  into 
attraction  as  a  fact  taught  by  experience.  ...  It  appeared 
to  me  that  if  all  bodies  act  "at  a  distance,"  in  proportion 
to  their  mass,  and  according  to  the  other  laws  which  we  know, 
such  enormous  masses  [the  mountains  of  Peru]  should  pro- 
duce a  marked  effect.  I  am  well  aware  that  they  are  very 
small  compared  with  the  whole  earth  ;  but  one  can  approach 
1000  or  2000  times  nearer  their  centre,  and  if  it  is  true  that 
c  33 


MEMOIRS    ON 

the  attractions  increase  not  only  simply  in  the  same  ratio  as 
the  distances  diminish,  but  in  the  inverse  ratio  of  their  squares, 
one  ought  to  have  a  kind  of  compensation. 

55.  I  shall  content  myself  with  justifying  this  in  the  case 
of  a  single  mountain  called  Chimborazo,  the  base  of  which 
one  is  obliged  to  pass  in  going  from  the  sea-side  at  Guaya- 
quil to  the  more  inhabited  part  of  the  province  of  Quito, 
which  is  enclosed  between  the  two  chains  of  mountains  here 
formed  by  the  Cordilleras,  whose  distance  apart  is  8  or  9 
leagues.  Chimborazo  must  be  3100  or  3200  toises  above  the 
sea-level  [he  afterwards  found  it  to  be  3217  toises],  and  1700 
or  1800  above  the  level  of  the  plateau.  We  know  exactly  the 
relative  heights  of  all  the  mountains  we  have  seen,  but  not 
having  yet  been  able  to  compare  any  one  with  sea-level,  we 
are  ignorant  of  their  absolute  heights.  Chimborazo  has  ro.ots 
which  extend  very  far  and  become  merged  in  those  of  the 
other  mountains,  so  that  it  is  very  difficult  to  determine  the 
true  extent  of  its  base.  It  must  be  more  than  10,000  or  12,000 
toises  in  diameter.  But  when  we  mount  as  high  as  possible, 
to  where  the  snow  begins,  which  is  850  toises  from  the  top 
and  renders  the  higher  parts  inaccessible,  the  mountain  is  still 
more  than  3500  toises  in  diameter.  The  top,  instead  of  ter- 
minating in  a  point,  is  rounded  and  blunt,  and  appears  from 
below  to  have  a  width  of  300  or  400  toises.  From  these  di- 
mensions one  can  estimate  its  huge  mass.  In  the  present 
investigation  we  need  to  know  its  height  above  ground  only, 
not  above  sea-level.  Even  so  it  must  be  20,000,000,000  cubic 
toises  in  volume.  This  is  about  the  y;oir;iro"oroooth  Part  OI1^J 
of  the  globe,  and  the  effect  of  the  attraction  would  be  absol- 
utely insensible,  if  one  considered  the  quantity  of  matter 
only.  But  as  we  can  place  ourselves  at  1700  or  1800  toises 
from  the  centre  of  gravity  of  the  mountain,  or  1900  times 
nearer  to  it  than  to  the  centre  of  the  earth,  this  proximity 
ought  to  increase  the  effect  about  3,600,000  times,  and  so 
make  it  about  2000  times  less  than  that  which  gravitation 
produces,  or  the  attraction  caused  by  the  whole  mass  of  the 
earth.  This  we  get  by  employing  only  a  rough  calculation 
and  the  lowest  estimates.  Calling  the  action  of  the  moun- 
tain 1,  and  that  of  the  earth  2000,  the  direction  of  attraction 
should  be  deflected  from  the  vertical  by  about  1'  43".  A 
plumb-line  which  would  be  directed  exactly  to  the  centre  of 

34 


THE    LAWS    OF    GRAVITATION 

the  earth,  if  its  mass  were  exposed  to  the  earth's  attraction 
a!6ne,  ought  then,  on  account  of  the  action  of  the  mountain, 
to  be  inclined  by  this  same  quantity,  which  is,  as  we  see,  quite 
considerable. 

56.  But    how   can    we   recognize    this    inclination;    for   all 
gravitating    bodies    must    be   equally    subject    to   it,  and    we 
seem  to  lack  a  term  of  comparison?     It  would  be  useless  to 
have  recourse  to   the  level   surfaces  of  the  heaviest  liquids, 
since   the    attraction    being    equally  altered   with   respect   to 
them,  their   surfaces,  instead    of    being   perfectly  horizontal, 
must  suffer  the  same  inclination.     We  see  plainly,  then,  that, 
in  order  to  judge  of  the  amount  of  this  alteration,  it  will  be 
of  no  use  to  look  just  about  us,  we  must  seek  another  ver- 
tical line   far    off    which    is   subject    to    no   action    from    the 
mountain.     But  again,  how  are  we  to   compare    one  vertical 
with   another ;    or    measure    the    angle   which    they  make  in 
meeting  towards  the  centre  of  the  earth,  and  that  with  suf- 
ficient accuracy?     If  while  on  the  mountain,  we  observe  with 
the  quadrant  the  height  of    a  point  far  off,  and  then  go  to 
that  point  and  measure  the  height  of  the  former  place,  it  is 
true  that  by  the  difference  of  these  two  heights  we  can  judge 
of  the  relative  positions  of  the  two  vertical  lines.     But    be- 
sides that  we  must  know  the  exact  distance  from  one  to  the 
other,  it  will  be  necessary  also  to  suppose  that  the  visual  ray 
is  a  straight  line  ;  and  it  is  not  only  certain  that  this  is  not 
true,  we  know  that  it  is  subject  by  refraction  to  a  very  ir- 
regular curvature.     We  cannot  determine  this  curvature  with 
sufficient  exactness  to  enable  us  to  find  the  effect  of  the  at- 
traction.    It  seems  to   me,  therefore,  that  we  must  seek  in 
the  heavens  a  term  of  comparison.     By  this  means,  however, 
we  shall  easily  overcome  every  difficulty;  and  what  a  moment 
ago  seemed  an  impossibility  becomes  at  once  very  simple. 

57.  We  have   but   to   station  ourselves   to   the  north  or  to 
the  south  of  a  mountain,  and  as  near  as  possible  to  its  centre 
of  gravity,  and  observe  the  latitude.     This  observation  can  be 
made  with    the  greatest    accuracy  only  by  using  a  quadrant 
or  other  equivalent  instrument  whose  plumb-line  will  be  de- 
flected toward  the  mountain;  this  is  the  same  as  saying  that 
the  zenith  will  recede   from    the   mountain.     Then  we   must 
go  east  or  west  of  this  station  to  such,  a  distance  that  the  at- 
traction is  negligible;  and  if  we  observe  the  latitude  in  this 

35 


MEMOIRS    ON 

second  place  with  the  same  care  and  with  the  same  means 
as  in  the  first,  it  is  evident  that  all  the  difference  which  we 
shall  observe  will  be  due  to  attraction.  In  order  to  have 
this  second  station  precisely  east  or  west  of  the  first,  we  must 
observe  the  azimuth  of  the  sun  at  its  rising  or  setting,  by 
finding  its  position  with  reference  to  some  easily  distinguished 
point  on  the  horizon  ;  in  doing  so  we  must  often  suppose  the 
latitude  known;  but  the  error  we  may  make  on  this  supposi- 
tion will  be  of  no  consequence,  and  it  will  always  be  easy  to 
find  two  stations  on  the  same  parallel  of  latitude  to  within 
3  or  4  sixtieths  of  a  second.  The  latitude  will  be  found  pre- 
cisely the  same  in  the  two  places,  if  the  vertical  line  has  not 
been  altered  in  the  first.  Suppose,  however,  that  without 
seeking  the  latitude,  we  observe  simply  the  meridian  alti- 
tudes of  a  star  at  the  two  stations ;  the  difference  of  these 
two  altitudes  will  indicate  equally  well  the  deflection  of  the 
vertical  line.  It  is  evident  that  all  the  stars  which  pass  the 
meridian  on  the  side  of  the  apparent  vertical  line  next  to  the 
mountain  will  appear  lower  at  the  first  station  than  at  the 
second  ;  for  as  the  plumb-line  approaches  the  mountain  the 
apparent  zenith  recedes  from  it  and  from  these  stars.  It  will 
be  quite  the  reverse  with  those  stars  which  pass  the  meridian 
on  the  other  side  of  the  apparent  vertical  line :  they  will  ap- 
pear higher  at  the  first  station.* 

58.  Instead  of  taking  the  stations  both  to  the  north  or  both 
to  the  south,  we  could  take  them  one  to  the  north  and  the 
other  to  the  south,  and  exactly  on  the  same  meridian  ;  then 
the  effect  of  the  attraction  would  be  doubled,  roughly  speak- 
ing, and  we  should  find  the  sum  of  the  contrary  attractions. 
The  vertical  line  would  be  inclined  in  opposite  directions  at 
the  two  stations;  and  the  altitudes  of  stars  which  would  be 
increased  in  the  one  would  be  decreased  in  the  other.  The 
physical  effect  being  doubled  would  be  more  sensible,  and 
more  susceptible  of  observation.  If  the  two  points  were 
equally  distant  from  the  centre  of  gravity  of  the  mountain, 
the  action  would  be  equal  at  both,  and  in  order  to  get  each 

*  \_This  method  of  doubling  the  deflection  caused  by  tlie  mountain,  by  observ- 
ing not  one  star,  but  at  least  two,  one  north  and  one  south  of  the  Rations, 
is  due  to  de  la  Condamine.  See  his  account  of  the  expedition  (8,  p.  68),  Zach 
(49)  and  Poynting  (185,  p.  14).  This  is  the  method  actually  employed  by 
Bouguer.] 


THE    LAWS    OF    GRAVITATION 

of  them  we  should  have  merely  to  take  half  of  the  quantity 
furnished  hy  the  comparison  of  the  observations.  In  other 
cases  the  division  would  be  a  little  more  difficult;  neverthe- 
less it  would  be  sufficient,  as  we  shall  shew  later,  to  divide 
the  sum  of  the  contrary  attractions  proportionally  to  the  pro- 
ducts of  the  quantity  by  which  each  station  is  more  north  or 
more  south,  respectively,  than  the  centre  of  gravity  of  the 
mountain  and  the  cube  of  the  distance  of  the  other  station, 
respectively,  from  the  same  centre.  Thus  we  are  under  the 
necessity  of  knowing  the  situation  of  each  station  with  refer- 
ence to  the  mountain;  but  we  must  know  the  distance  from 
one  station  to  the  other  also,  in  order  to  determine  geometric- 
ally the  difference  of  latitude  between  them.  It  is  evident 
that  this  difference  must  itself  produce  a  change  in  the  alti- 
tude of  each  star,  and  we  must  know  it  before  we  can  tell  what 
is  the  double  effect  of  the  attraction.  To  obtain  the  difference 
in  latitude  of  the  two  places,  it  would  suffice  ordinarily  to 
measure  to  the  east  or  to  the  west  of  the  mountain  a  base 
directed  nearly  north  and  south,  and  to  form  on  this  base  two 
triangles  which  end  at  the  two  stations. 

59.  This   way  of  making   two  observations   from    different 
sides  of  the  same  mountain  in  order  to  render  the  effect  of  the 
attraction  more  sensible,  seems  to  me  the  more  useful  method, 
as  it  depends  less  on  the  peculiarities  of  the  places.     We  can 
sometimes  double  the  effect  also  by  making  the  first  observa- 
tion at  the   north   of  one   mountain   and    the  second  at  the 
south  of  another.     If  the  two  stations  are  not  exactly  on  the 
same  east  and  west  line,  we  have  only  to  determine  geometric- 
ally their  difference  of  latitude,  and  take  account  of  it  in  the 
comparison  of  the  altitudes  of  the  stars. 

60.  Finally,  it  is  not  only  by  observations  made  at  the  north 
or  at  the  south  that  we  can  discover  whether  mountains  are 
capable  of  acting  "at  a  distance"  ;  it  can  be  done  also  by  ob- 
servations made  at  the  east  or  the  west;   but  with  this  differ- 
ence, that  it  will  be  no  longer  a  question  of  observing  latitude, 
or  of  taking  the  meridian  altitudes  of  stars;  it  will  be  only  a 
question  of  determining  time  exactly.     It  appears  to  me  that 
this  last  method  would  be  often  preferable  to  the  preceding 
ones,  except  that  it  requires  two  observers.     Suppose  that  the 
first  of  these  is  on  the  east  side  of  a  mountain,  and  the  second 
on  the  west  side  of  another,  or  of  the  same,  mountain.     If  each 

37 


MEMOIRS    ON 

of  them  regulates  carefully  a  chronometer  by  corresponding  al- 
titudes, it  is  evident  that  all  these  altitudes  being  altered  by 
the  attraction  which  deflects  the  plumb-line,  each  chronome- 
ter will  be  regulated  as  if  the  meridian  were  not  exactly  vertic- 
al, but  inclined  below  toward  the  mountain,  and  above  away 
from  it.  Let  us  suppose  that  the  attraction  amounts  to  a  min- 
ute of  arc,  and  that  the  two  mountains  are  on  the  equator  ; 
the  first  chronometer  will  denote  midday  4  seconds  of  time 
too  soon,  and  the  other  4  seconds  too  late.  Thus,  neglect- 
ing the  difference  of  longitude,  which  we  could  easily  find  by 
measuring  trigonometrically  the  distance  of  the  two  observ- 
ers apart  and  reducing  this  distance  to  degrees  and  min- 
utes, there  would  be  a  difference  of  8  seconds,  of  time  be- 
tween the  two  chronometers.  If  the  two  mountains  instead 
of  being  on  the  equator  were  at  latitude  60°,  each  minute  of 
inclination  which  the  attraction  produced  in  a  plumb-line 
would  produce  8  seconds  of  difference  in  the  time  of  mid- 
day, and  therefore  16  seconds  difference  in  the  chronometers. 
Finally,  to  judge  of  the  attraction  we  need  only  know  the  exact 
difference  between  the  chronometers;  and  to  find  this,  it  would 
always  be  sufficient  to  agree  upon  a  signal,  by  fire  or  other- 
wise ;  and  to  observe  at  both  stations  the  minute  and  second 
of  the  instantaneous  appearance  of  this  signal. 

61.  I  return  to  the  first  method  because  it  appears  to  me  to 
be  the  simplest ;  that  is,  suppose  we  station  ourselves  always 
to  the  north  or  to  the  south  of  the  mountain  and  confine  our- 
selves to  observations  of  the  latitude.  It  is  evident  that  if. we 
take  at  each  station  the  meridian  altitude  of  one  star  only, 
we  must  know  to  the  last  degree  of  nicety  the  condition  of  the 
quadrant  we  are  using.  There  is  no  lack  of  methods  for  veri- 
fying this  instrument,  but  there  is  one  which  is  extremely 
valuable  in  the  present  instance,  because,  at  the  same  time  as 
we  work  at  verifying  the  quadrant,  we  are  making  the  observ- 
ations which  decide  the  question  at  issue  ;  and  in  thus  abridg- 
ing the  operations  we  avoid  opportunities  for  errors.  This 
method  is  to  take  the  meridian  altitudes  of  an  equal  number 
of  stars  toward  the  north  and  toward  the  south,  and,  provided 
that  the  state  of  the  instrument  does  not  vary  from  one  observ- 
ation to  another,  it  does  not  matter  if  it  does  change  from 
day  to  day.  If  it  makes  the  altitudes  of  the  stars  on  one  side 
the  zenith  too  great,  it  will  produce  the  same  effect  with  re- 

38 


THE    LAWS    OF    GRAVITATION 

spect  to  those  on  the  other  side.  Thus  the  change  will  influ- 
ence only  the  sum  of  the  altitudes  or  the  complements  of  the 
altitudes,  and  will  not  alter  the  difference  of  the  altitudes  taken 
on  the  different  sides.  The  attraction,  on  the  contrary,  will  not 
alter  the  sum,  but  will  change  the  difference;  because  at  the 
same  time  that  it  makes  the  stars  on  one  side  too  high,  it  makes 
those  on  the  other  side  too  low.  It  will  always  be  easy  to  sep- 
arate these  two  causes,  and  we  shall  not  attribute  to  the  one 
that  which  arises  from  the  other.  To  obtain  at  one  stroke  the 
effect  of  attraction  without  being  obliged  to  know  the  state  of 
the  quadrant  or  the  declinations  of  the  stars,  we  need  only  ex- 
amine whether  the  differences  of  the  meridian  altitudes  taken 
towards  the  north  and  towards  the  south  are  the  same  at  the 
two  stations,  or  whether  they  are  subject  to  a  second  differ- 
ence. But  it  is  necessary  to  remark  that  the  altitudes  being 
increased  on  the  one  side  while  they  are  diminished  on  the 
other,  it  is  the  half  of  this  second  difference  which  denotes 
the  physical  effect  of  the  attraction,  both  when  this  effect  is 
single  and  when  it  is  double.  In  this  latter  case,  it  will  be 
necessary  to  divide  the  total  effect  in  the  ratio  which  the  sep- 
arate effects  ought  to  have. 

[Bouguer  then  proves  this  ratio  to  be  that  mentioned  above  (p. 
37).  He  admits  that  some  mountains  might  sliew  less  attraction 
than  that  required  by  Neivtotis  law  (or  even  none),  due  to  the 
existence  of  great. cavities  in  the  mass.  He  discusses  the  different 
mountains  in  the  neighbourhood  of  Quito,  and  for  various  reasons 
decides  upon  Chimborazo  as  the  one  most  suitable  for  the  experi- 
ment.] 

EXAMINATION  OF  THE  ATTRACTION  OF  CHIMBORAZO 

65.  I  did  not  ascend  this  mountain  alone  as  I  did  the  pre- 
ceding one.  I  had  some  time  before  communicated  my  design 
and  all  my  views  to  M.  de  la  Condamine,  and  when  on  the 
point  of  carrying  them  out  I  mentioned  them  to  M.  de  Ulloa, 
one  of  the  two  naval  lieutenants  who  had  assisted  in  the  ob- 
servations both  of  myself  and  of  M.  de  la  Condamine  ever  since 
our  arrival  in  the  domains  of  His  Catholic  Majesty.  These 
gentlemen  obligingly  offered  to  accompany  me,  not  only  in  the 
preparatory  examination,  but  also  during  the  stay  it  was  necess- 
ary to  make  on  the  mountain  side  ;  and  as  I  knew  it  would  be 
to  the  advantage  of  the  observations,  I  hastened  to  accept  the 


MEMOIRS    ON 


offer.  I  had  already  thought  that  Chimborazo  fulfilled  ap- 
proximately the  necessary  conditions  :  I  knew  that  it  was  very 
easy  of  access ;  it  could  be  seen  from  Quito,  or  rather  from 
Pichincha,  from  which  it  was  75,000  toises  distant;  and  I  had 
already  measured  its  height.  ...  On  December  4th  we  estab- 
lished ourselves  on  the  south  side  of  the  mountain,  at  the  bot- 
tom of  the  snow  line,  829  toises  below  the  summit,  but  about 
2400  above  sea-level,  and  exactly  910  toises  above  the  place  at 
Quito  where  I  have  always  made  my  observations,  and  344 
toises  above  that  part  of  Pichincha  where  there  is  a  cross 
which  can  be  seen  from  all  parts  of  the  city,  and  where  I  passed 
some  days  in  March,  1737,  in  order  to  observe  the  astronomical 
refraction.  I  shall  not  speak  of  the  cold  and  the  other  discom- 
forts we  had  to  put  up  with  ;  snow  covered  our  tent  and  all 
the  ground  around  as  far  as  800  or  900  toises  below  us,  and  we 
lived  in  fear  of  being  buried  under  its  weight.  It  needed  con- 
tinual vigilance  in  order  to  avoid  it.  [M.  de  Ulloa  fell  ill, <  and 
had  to  descend  the  mountain  on  December  15th.] 

[Left  alone,  Bouguer  and  de  la  Condamine  observed  the  alti- 
tudes of  10  stars,  4  on  the  south  side  and  6  on  the  north.  Tlie 
following  are  the  altitudes  as  affected  by  the  error  of  the  instru- 
ment and  by  refraction;  they  are  the  means  of  the  readings  of  the 
two  observers.] 

67. 


On  the  north  side  : 
Capella 

Meridian  altitudes  at  the  first  station 

On  14th  Dec. 

On  15th  Dec. 

0           '            " 

42    50    42| 

59    53    55 
66     18     15 
69      0    20 
72    33    42£ 

32    58    10 

38    58    52i 
72      6    36| 
75      9     12^ 

o         '          " 

42    50    30 
56      6      5 
59    53    57£ 
66    18    50 
69      0    17£ 
72    33    57£ 

32    58    25 
38    58    55 

72      6    35 
75      9    40 

First  head  of  Gemini                           .    . 

Second  head  of  Gemini.   .           

Second  horn  of  Aries 

First  horn  of  Aries 

Aldebaran  

On  the  south  side  : 
Acarnar 

Canopus                         ....           .    . 

Tail  of  Cetus 

Sirius 

68.  We  observed  the  meridian  altitude  of  the  sun  three 
times.  De  la  Condamine  found  it  at  the  lower  edge  on  De- 
cember 15th  to  be  67°  54'  26".  This  altitude,  which  is  cor- 
rected for  the  error  of  the  instrument,  but  not  for  refraction 

40 


THE    LAWS    OF    GRAVITATION 

and  parallax,  gives  1°  2$'  53"  south  for  the  latitude  of  the 
place  where  we  were.  I  observed  it  on  the  5th  and  12th  ;  on 
the  5th  we  had  not  yet  regulated  the  chronometer  nor  traced 
the  meridian,  and  I  found  1°  30'  16".  On  the  12th  I  observed 
the  apparent  altitude  of  the  lower  edge  of  the  sun  to  be 
68°  5'  34" ;  which  gives  1°  30'  6". 

69.  As  soon  as  we  were  established  on  Chimborazo,  I  had  sent 
a  tent  about  a  league  and  a  half  to  the  west  to  a  place  called 
I'Arenal  to  serve  as  the  second  station.  [Bouguer  then  describes 
the  measurements  made  to  find  the  exact  position  of  the  second 
station;  it  ivas  3570  toises  distant  from  the  first,  174  toises  lower, 
and  somewhat  south  of  ivest  of  it.  They  began  their  observations 
from  the  second  station  on  Dec.  16th.  Here  they  suffered  more 
from  the  wind  and  cold  than  at  the  more  elevated  station,  as  they 
were  more  exposed  to  the  prevailing  east  wind.  It  filled  their 
eyes  with  dust  and  continually  threatened  to  overturn  the  tents.* 
Tile  screws  of  the  quadrant  could  not  be  turned  at  night  without 
applying  heat  to  them.  Then  follows  a  table  of  the  altitudes  as 
observed.  Since  the  second  station  was  505  toises,  or  32",  south 
of  the  first,  we  must  increase  by  32"  all  the  altitudes  of  stars  ob- 
served toward  the  north,  and  diminish  the  others  by  the  same 
amount.  Moreover,  since  the  second  station  was  lower  than  the 
first  by  174  toises,  the  altitudes  observed  at  the  second  station 
must  be  diminished,  to  reduce  them  to  the  level  of  the  first,  on  ac- 
count of  the  excess  of  astronomical  refraction.  The  following 
are  the  corrected  altitudes,  and  the  differences  for  each  star  of 
the  mean  determinations  at  the  two  stations  :] 


*  [For  de  la  Condamine's  account  of  the  experiments,  see  his  Journal  (8, 
p.  69  and  8i,  pt.  2,  p.  146).] 

41 


MEMOIRS    ON 


CORRECTED  ALTITUDES  AT  THE  SECOND  STATION 

Excess  of  altitudes 
at  the  first  station 
over  those  at  the 
second,   after   the 
latter    have   bee" 
corrected 

On  21st  Dec. 

On  22d  Dec. 

On  the  north  side: 

O           /           " 

42  49  10 

66   16  59 

68  59     6 
72  32  31! 

32  56  53 

38  57   16 

72     4  47| 
75     8     0 

0            /            'I 

42  49   15 

59  53  23| 
66  17  16 
68  59   11 
72  32  36! 

32  56  28 

38  57  36 
72     4  50 

75     8     7! 

i     n 

1  24 

1  25 
1   10 
1     6* 

1   37! 
1   28 

1   48 
1   22! 

First  head  of  Gemini... 
Second  head  of  Gemini. 
Second  horn  of  Aries.  .  . 
First  horn  of  Aries  
Aldebaran 

On  the  south  side: 
Acarnar  

Canopus 

Tail  of  Cetus  .  . 

Sirius  

[Bouguer  considers  the  observations  on  the  tail  of  Cetus  and 
the  first  horn  of  Aries  as  the  best,  but  thinks  it  most  legitimate 
to  take  the  mean  of  the  altitudes  of  each  star  at  each  station  and 
to  give  equal  weight  to  their  differences.  These  differences  are 
given  in  the  last  column  of  the  preceding  table,  and  are,  he  main- 
tains, too  large  and  too  uniform  to  be  due  to  any  defect  in  the  ob- 
servations. The  averages  of  the  excess  of  all  the  stars  on  the 
north  side  and  all  those  on  the  south  side  are  now  to  be  taken.} 

74.  ...  They  give  about  1'  19"  as  the  mean  excess  for  the 
north  stars,  and  1'  34"  for  the  south.  The  second  difference  is 
15".  I  leave  it  to  my  readers  to  say  whether  such  a  quantity 
is  sufficiently  established  by  the  means  employed.  My  quad- 
rant was  2.5  ft.  in  radius,  and  it  must  be  remarked  that  any 
errors  which  may  exist  in  its  graduation  are  of  no  importance 
here  ;  since  we  have  to  do  not  with  the  altitudes  themselves, 
but  with  their  differences.  Suppose  we  admit  the  15",  it  will 
give  7".5  for  the  effect  of  attraction;  it  would  be  much  greater 
if  we  compared  the  tail  of  Cetus  with  the  first  horn  of  Aries. 
However  this  is  not  the  complete  and  absolute  effect;  for  if 
attraction  really  takes  place,  the  mountain  must  have  some 
effect  at  the  second  station,  which  was  about  4572  toises  from 
the  centre  of  the  mountain,  and  61°. 5  to  the  west  of  south. 
At  the  first  station  we  were  nearly  16°  west  of  south,  and  1753 


*  [This  is  evidently  a  misprint  for  1'  16".] 
42 


THE    LAWS    OF    GRAVITATION 

toises  distant.  From  these  data  we  find  that  the  effect  at  the 
nearer  station  is  to  the  effect  we  ought  to  find  at  the  other  as 
1358  : 100,  or  as  13^  :  1  nearly.  But  since  our  observations 
give  only  the  difference  of  the  two  effects,  we  must  increase 
7". 5  by  a  13th  or  14th  part  of  itself  in  order  to  have  the  total 
effect  [which  makes  it  8"]. 

75.  We  must  admit  that  this  effect  is  very  different  from 
what  we  had  expected.  But  we  know  so  little  about  the 
earth's  density,  and  on  the  other  hand  that  of  the  mountain 
may  be  so  different  from  that  which  we  have  assumed  it  to  be, 
that  there  is  no  reason  to  be  surprised  at  anything.*  [It  is  a 
tradition  among  the  natives  that  Chimborazo  is  an  extinct  vol- 
cano, and,  if  so,  its  density  would  be  very  hard  to  estimate. 
Houyuer  thinks  it  might  be  better  to  experiment  on  smaller  and 
denser  mountains.]  It  is  very  probable  that  we  shall  find  in 
France  or  in  England  some  hill  of  sufficient  size,  especially  if 
we  double  the  effect ;  and  I  shall  be  delighted  if  I  find  on  my 
return  that  the  experiments  that  shall  have  been  made  either 
confirm  mine  or  throw  new  light  on  the  matter.  [At  Riobamba 
in  Peru,  December  30,  1738.] 

[In  an  appendix  Bouguer  states  that  after  a  more  thorough 
survey  of  the  Cordilleras  he  failed  to  find  a  more  satisfactory 
place  at  which  to  repeat  his  experiment.  He  suggests  that  the 
converse  effect  be  experimented  upon  :  viz.,  the  decrease  in  grav- 
ity due  to  some  deep  canon  among  mountains.  Assuming  such 
great  cavities  in  Chimborazo  as  ivould  make  its  real  only  half 
its  apparent  volume,  he  finds  his  results  would  make  it  6  or 
7  times \  less  dense  than  the  earth;  this  he  thinks  not  unreason- 

*  \J)e  la  Condamine  also  laid  little  stress  on  the  numerical  remit  of  the  plumb- 
line  experiment ;  for  he  says  (8,  p.  69),  "ifwe  can  deduce  from  it  nothing 
decisively  in  favour  of  the  Newtonian  attraction,  at  least  we  find  nothing  con- 
trary to  that  theory."] 

f  [In  a  critical  analysis  of  all  the  experiments  made  to  determine  the  density 
of  the  earth  up  to  that  time,  Saigey  (74,  p.  151  and  in  his  "Petite  Physique  du 
Globe,"  Paris,  1842,  pt.  2,  p.  151),  in  1842,  stated  that  Bouguer's  calculations 
were  erroneous,  because  he  confused  the  centre  of  attraction  with  tJie  centre  of 
gravity  of  the  mountain;  he  refers  to  a  metliod,  by  means  of  which,  using 
Bouguer 's  own  mean  result,  he  deduced  the  density  of  the  earth  to  be  4.62 
times  that  of  tlie  mountain  ;  but  adds  that  if  he  had  used  only  the  results  from 
observations  on  the  tail  of  Cetus  and  the  first  horn  of  Aries,  tJie  result  would 
have  been  1.83,  which  is  almost  exactly  the  same  as  that  found  by  Maskelyne  by 
the  same  method  for  the  hill  Schehallien.~\ 

43 


MEMOIRS    ON    THE    LAWS    OF    GRAVITATION 

able.  Bouguer  further  remarks  that  the  distribution  of  density 
in  the  earth  may  be  such  that  the  maximum  attraction  of  the 
earth  is  not  at  its  surface  but  at  some  distance  beneath  it.] 


[In  connection  with  this  work  of  Bouguer  should  be  read  a 
paper  (9)  presented  by  him  to  the  Academy  on  April  28,  1756, 
on  the  possibility  of  detecting  the  deflection  of  a  phimb-line  due 
to  the  ebb  andfloiv  of  the  tide. 

Valuable  accounts  and  discussions  of  Bouguer's  work  in  Peril 
are  given  by  von  Zach  (43,  44,  and  49),  Schmidt  (G4,  vol.  2, 
p.  475),  Todhunter  (140,  chap.  12),  Zanotti- Bianco  (148£,  jo/. 
2,  pp.  122-25),  and  Poynting  (185,  pp.  10-14.] 


PIERRE  BOUGUER  was  born  in  1698  at  Croisic,  Bretagne,  and 
was  educated  at  the  Jesuit  College  at  Vannes.  He  succeeded 
his  father  as  professor  of  hydrography  at  Croisic  in  1713,  and 
in  1730  accepted  a  similar  position  at  Havre.  His  investiga- 
tions concerning  the  intensity  of  light,  embodied  in  a  work, 
Essai  d'optique  sur  la  gradation  de  la  lumiere,  published  in 
1729,  led  to  his  being  elected  a  member  of  the  Academy  of 
Sciences  in  1731 ;  he  was  promoted  to  the  office  of  pensioned 
astronomer  in  1735.  Along  with  two  other  members  of  the 
Academy,  MM.  de  la  Condamine  and  Godin,  he  was  sent  to 
Peru  in  1735  to  measure  the  length  of  an  arc  of  the  meridian 
near  the  equator.  Their  labors  there  lasted  ten  years,  and  the 
results  of  their  observations  were  published  in  1749  in  La 
Figure  de  la  Terre,  from  which  we  have  given  the  preceding  ex- 
tracts. For  several  years  afterwards  Bouguer  was  engaged  in 
a  bitter  controversy  witli  de  la  Condamine  concerning  their  re- 
spective shares  in  the  Peruvian  researches.  In  addition  to  his 
works  on  photometry,  he  published  several  valuable  treatises 
on  navigation,  and  various  papers  on  atmospheric  refraction 
and  other  optical  problems,  and  on  mechanics.  He  died  at 
Paris  in  1758. 

44 


THE    BEETIER   CONTEOVEESY 


45 


THE    BERTIEB  CONTROVERSY 


IN  June,  1769,  there  appeared  in  the  Journal  des  Sciences  et 
des  Beaux  Arts  a  letter  (11)  by  a  M.  Coultand,  who  signed 
himself  Former  Professor  of  Physics  at  Turin.  In  it  he  de- 
scribed some  pendulum  experiments  made  in  the  Alps  of  Savoy. 
He  claimed  to  have  found  that  at  a  height  of  1085  toises  above 
the  base  of  the  mountain  the  pendulum  gained  28'  in  2  months ; 
20'  22"  in  3  months  at  a  height  of  514  toises;  and  15'  4"  in  175 
days  at  a  height  of  210  toises.  So  that  it  appeared  as  if  the 
attraction  of  gravitation  increased  with  the  distance  from  the 
earth's  centre,  instead  of  behaving  according  to  the  Newtonian 
law.  A  full  account  of  the  apparatus  and  observations  was 
added,  and  there  seemed  no  reason  why  credence  should  not  be 
given  to  the  results.  The  advocates  of  the  Newtonian  theory 
felt  called  upon  to  account  for  this  phenomenon  consistent- 
ly with  their  doctrine.  D'Alembert  (12  and  13,  vol.  6,  pp. 
85-92)  attacked  the  problem  and  found  that  the  Newtonian 
theory  was  adequate  to  explain  the  fact,  provided  the  mean 
density  of  the  earth  were  about  three  eighths  of  that  of  the 
mountain.*  An  abstract  of  Goultaud's  alleged  observations  is 
given  by  David  (14). 

In  Dec.,  1771,  another  letter  appeared  in  the  same  journal 
(15)  signed  by  one  Mercier,  and  addressed  to  Gessner,  Pro- 
fessor of  Physics  in  the  Univ.  of  Geneva.  •  It  described  experi- 
ments made  in  Valois  similar  to  those  of  Coultaud  and  with 
similar  results,  namely,  that  the  attraction  of  gravitation  is 
directly  proportional  to  the  square  of  the  distance.  D'Alem- 
bert  then  discussed  the  question  again  (13,  vol.  6,  pp.  93-98). 
Further  explanations  on  the  Newtonian  theory  were  forth- 

*  Compare  the  conclusion  of  Bonguer  on  p.  31  of  this  volume. 

47' 


MEMOIRS    ON 

coming  from  Le  Sage  (16)  and  Lalande  (17).  Roiffe  also  dis- 
cussed the  experiments  (18). 

These  results  of  Coultaud  and  Mercier  seem,  however,  to 
have  been  a  cause  of  great  exultation  to  a  certain  number  of 
scientists,  especially  ecclesiastics,  who  contended  that  the  New- 
tonians wished  to  take  from  them  their  Father,  their  God,  in 
asserting  that  bodies  attract  and  move  of  themselves  without 
any  Prime  Mover.  It  is  hard  to  believe  that  this  feeling 
existed  so  late  as  a  century  ago.  One  of  the  most  active  of 
the  opponents  of  the  theory  of  Newton  was  Father  Bertier, 
de  1'Oratoire,  who  founded  his  4th  volume  of  Les  Principes 
Physiques  on  the  above  experiments.  For  several  years  a  warm 
discussion  raged  among  French  physicists  over  the  question. 

Le  Sage,  having  had  his  suspicions  aroused  by  some  passage 
in  Mercier's  letter,  began  a  careful  investigation  into  the  gen- 
uineness of  the  experiments  both  of  Coultaud  and  Mercier. 
He  found  them  to  be  fabrications  from  beginning  to  end  (19). 
Le  Sage  does  not  mention  whom  he  supposes  to  be  the  per- 
petrators or  instigators  of  the  fraud. 

A  new  impetus  was  given  to  the  discussion  by  the  publica- 
tion of  2  letters  (20)  from  Father  Bertier  describing  experi- 
ments with  the  balance  similar  to  those  performed  by  members 
of  the  Royal  Society  of  London  a  century  before  ;  but  Bertier 
writes  as  if  the  idea  were  entirely  a  new  one.  The  length  of 
the  string  used  to  suspend  the  weight  from  one  arm  of  the 
balance,  after  it  had  been  counterpoised  in  the  pan  above,  was 
74  ft.  In  one  case  weights  of  25  Ibs.  were  used,  and  when  one 
of  them  was  suspended  at  the  end  of  the  string  it  lost  in  weight 
1  ounce  3.5  drachms.  Bertier  concluded,  much  to  his  satisfac- 
tion, that  bodies  weigh  more  the  farther  they  are  from  the 
centre  of  the  earth.  Roiffe  followed  with  a  paper  (21)  discuss- 
ing the  experiments  made  thus  far  and  remarking  that  Bertier 
had  not  taken  account  of  the  difference  in  the  density  of  the 
air  at  the  two  levels.  Le  Sage  also  criticised  Bertier  very 
harshly  (22).  Repetitions  of  Bertier's  experiment  were  made 
by  M.  David,  and  Fathers  Cotte  and  Bertier  (23),  the  one  with 
a  string  of  170  ft.  and  weights  of  1220  Ibs.,  the  others  with 
a  string  of  45  ft.  and  weights  of  150  Ibs.  The  one  reported  a 
loss  in  weight  of  1  oz.,  the  others  of  2  Ibs.,  in  the  same  direc- 
tion as  indicated  by  Bertier's  first  experiment.  An  article  by 
David  (24)  in  answer  to  Le  Sage  contains  some  scornful  strict- 

48 


THE    LAWS    OF    GRAVITATION 

ures  on  Newton  and  his  principles  which  form  amusing  read- 
ing at  this  late  date.  Rozier  (25)  criticised  all  the  experi- 
ments made  on  the  laws  of  gravitation  ;  he  refers  to  some 
more  made  by  Bertier  (26)  from  which  the  latter  concluded 
that  the  loss  in  weight  was  proportional  to  the  length  of  string 
and  to  the  weight.  Rozier  then  announced  the  details  of  some 
experiments  of  a  similar  kind  made  by  himself,  which  gave 
quite  discordant  results.  David  wrote  another  letter  (27)  with 
more  details  of  his  experiments,  but  adding  nothing  of  value. 
Bertier  followed  with  a  similar  letter  (28).  A  committee  of 
the  Academy  of  Dijon  repeated  the  experiments  with  a  sensit- 
ive balance,  and  found  (29)  no  change  in  weight  except  that 
due  to  the  different  densities  of  the  air  at  the  higher  and 
lower  levels.  Some  experiments  on  this  same  subject  were 
made  by  Achard  (33)  ;  he  found  by  using  first  a  string  and 
then  a  brass  chain  with  which  to  suspend  the  masses,  that  the 
changes  in  weight  of  the  suspended  mass  could  be  ascribed  to 
variations  in  the  temperature  and  dampness  of  the  air.  Dolo- 
mieu  (30)  made  some  experiments  with  a  weight  suspended  in 
a  mine,  similar  to  those  of  Dr.  Power  (page  2) ;  his  results 
permitted  no  definite  conclusions  as  to  change  in  weight.  In 
connection  with  this  work  we  might  notice  a  valuable  article 
by  Le  Sage  (34)  on  the  history  of  the  theory  of  gravitation  and 
the  experiments  made  concerning  it.  He  gives  a  brief  account 
of  the  views  of  Gilbert,  Bacon,  Kepler,  Beaugrand,  Fermat, 
Pascal,  Roberval,  Descartes,  and  Gassendi  ;  finally  he  dis- 
cusses Dolomieu's  experiments  and  the  possible  variation  in 
density  beneath  the  surface  of  the  earth. 

The  controversy  closes  with  the  appearance  of  a  letter  (36) 
from  Bertier  making  an  humble  retraction  of  his  statements 
regarding  the  deductions  to  be  drawn  from  his  experiment ;  he 
admits  that  such  experiments  do  not  prove  that  bodies  weigh 
more  as  they  are  farther  from  the  earth,  but  he  declines  to  give 
up  his  belief  that  such  is  the  case.  He  again  inveighs  against 
those  who  "  by  means  of  101  different  laws,  which  they  make 
God  create  to  cover  their  ignorance,  explain  everything  with  a 
facility  that  is  truly  delightful." 
D  49 


THE   SCHEHALLIEN    EXPEEIMENT 


51 


THE   SCHEHALLTEN   EXPEEIMENT 


IN  1772,  Maskelyne,  the  English  Astronomer  Royal,  pro- 
posed to  the  Royal  Society  (31)  that  the  experiment  of  Bou- 
guer  on  the  attraction  of  a  mountain  be  repeated  in  Great 
Britain,  as  Bongner  himself  had  suggested  30  years  before. 
Maskelyne  had  been  informed  of  two  places  which  might  be 
convenient  for  the  purpose.  One  was  near  the  confines  of 
Yorkshire  and  Lancashire,  on  the  hill  Whernside  ;  the  other 
in  Cumberland,  on  the  hill  Helvellyn.  The  proposal  was 
favourably  received  by  the  Society,  and  Mr.  Charles  Mason  was 
sent  to  examine  various  hills  in  England  and  Scotland,  and  to 
select  the  most  suitable  (32).  Mason  found  that  the  two  hills 
referred  to  by  Maskelyne  were  not  suitable;  and  fixed  upon 
Schehallien  in  Perthshire  as  offering  the  best  situation.  At 
the  earnest  solicitation  of  the  Royal  Society,  Maskelyne  him- 
self undertook  to  make  the  necessary  observations.  He  had  at 
his  disposal  a  10-foot  zenith  sector,  and  all  his  other  instru- 
ments were  the  best  of  their  kind  at  the  time.  The  work  was 
begun  in  the  summer  of  1774.  The  method  of  finding  the  de- 
flection of  the  plumb-line  due  to  the  hill  was  exactly  the  second 
of  the  methods  described  by  Bouguer  (page  36) ;  he  took  read- 
ings of  the  zenith-distances  of  certain  stars  at  two  stations,  one 
north  and  one  south  of  the  hill,  and  by  this  means  doubled  the 
deflection  of  the  plumb-line.  Between  June  30th  and  Septem- 
ber 22d  he  took  1G9  star  observations  from  the  south  station, 
and  1G8  from  the  north  station ;  in  all  337  observations  on  43 
stars.  At  the  same  time  a  very  elaborate  survey  by  triangula- 
tion  was  made  of  the  dimensions  and  form  of  the  hill.  This 
was  considered  as  made  up  of  a  very  large  number  of  prisms, 
sufficient  data  for  the  determination  of  each  of  which  were  col- 
lected during  the  survey. 

53 


MEMOIRS    ON 

In  his  paper  (32)  describing  the  operations,  Maskelyne  cal- 
culates, from  40  only  out  of  the  337  observations,  that  the  ap- 
parent difference  of  latitude  between  the  two  stations  is  54".6.* 
The  true  difference  of  latitude  is  43",  leaving  11". 6  due  to  the 
contrary  attractions  of  the  hill. 

From  a  rough  calculation,  assuming  the  density  of  the 
mountain  to  be  the  same  as  the  mean  density  of  the  earth, 
and  that  the  law  of  attraction  is  that  of  the  inverse  square 
of  the  distance,  Maskelyne  found  that  the  attraction  should  be 
twice  that  found  by  observation.  Hence  the  mean  density  of 
the  earth  is  twice  that  of  the  hill.  A  more  exact  calculation 
was  promised  for  the  future.  Maskelyne  draws  two  main  con- 
clusions: (1),  that  Schehallien  has  an  attraction,  and  so,  there- 
fore, has  every  mountain ;  (2),  that  the  inverse  square  law  of 
the  distance  is  confirmed  ;  for  if  the  force  were  only  a  little 
affected  by  the  distance,  the  attraction  of  the  hill  would  be 
wholly  insensible. 

The  survey  of  the  hill  and  its  environs  was  made  during  the 
years  1774,  1775  and  1776.  The  calculation  of  the  attraction 
of  the  hill  from  these  measurements  was  undertaken  by  Hut- 
ton,!  who  employed  several  new  and  interesting  methods.  A 
full  account  will  be  found  in  his  paper  (37  and  47,  vol.  2, 
pp.  1-68).  Assuming  that  the  density  of  the  hill  is  the  same 
as  the  mean  density  of  the  earth,  Hutton  found  that  the  attrac- 
tion of  the  earth  is  to  the  sum  of  the  contrary  attractions  of 
the  hill  as  9933  : 1.  Now  Maskelyne  had  found  the  deflection 
due  to  the  contrary  attractions  of  the  hill  to  be  11". 6 ;  whence 
the  attraction  of  the  earth  is  to  the  sum  of  the  attractions  of 
the  hill  as  1  :  tan.  11". 6,  or  as  17781  :  1  ;  or,  allowing  for  the 

*  Von  Zacli  (49,  App.,  pp.  686-692)  has  calculated  the  results  from  all 
of  the  337  observations,  and  finds  for  the  apparent  difference  of  latitude 
54". 651,  and  for  the  deflection  due  to  the  contrary  attractions  of  the  hill 
11". 632  ;  which  is  in  entire  accord  with  Maskelyne's  calculations.  Saigey 
(74,  p.  153)  also  subjected  the  result  to  a  test  which  was  satisfactory. 
Zanotti-Bmnco  states  (148£,  pt.  2,  p.  134)  that  Saigey  maintained  that 
Maskelyne  did  not  choose  his  station  at  the  most  favourable  part  of  the 
hill-side,  and  that  if  he  had  done  so  he  would  have  found  the  deflection 
14"  instead  of  11". 6. 

f  For  Button's  own  estimate  of  his  share  in  the  work,  and  for  his  con- 
tempt for  Cavendish's  experiment,  see  bibl.  No.  45.  For  a  good  account 
of  Hutton's  method  of  calculation,  see  Zanotti-Bianco  (148^,  pt.  2,  pp.  126- 
32);  see  also  Helmert  (148,  vol.  2,  pp.  368-80). 

54 


THE    LAWS    OF    GRAVITATION 

centrifugal  force,  as  17804  :  1  nearly.  Hence  the  mean  density 
of  the  earth  is  to  the  density  of  the  hill  as  17804  :  9933,  or  as 
9  :  5  nearly.  Assuming  the  specific  gravity  of  the  hill  to  be 
about  2.5,  Hutton  remarks  that  this  would  give  4.5  as  the 
mean  specific  gravity  of  the  earth.  Hutton  revised  this  result 
in  his  "Tracts"  (47,  vol.  2,  p.  64);  he  takes  the  specific 
gravity  of  the  hill  as  3,  and  hence  the  specific  gravity  of  the 
earth  would  be  5.4  nearly. 

Playfair,  with  the  aid  of  Lord  Webb  Seymour,  made  a  care- 
ful lithological  survey  of  Schehallien,  and  published  his  re- 
sults in  1811  (46  and  48).  He  found  that  the  hill  was  made 
up  of  two  classes  of  rock,  quartz  of  specific  gravity  2.639876, 
and  micaceous  rock,  including  calcareous,  of  specific  gravity 
2.81039.  From  two  suppositions  as  to  the  distribution  of  these 
two  components  in  the  interior  of  the  hill,  using  Hutton's  data 
for  the  attraction,  Playfair  calculated  the  mean  density  of  the 
earth  to  be  4.55886  and  4.866997  respectively.  Playfair  con- 
sidered the  experiment  on  Schehallien  so  exact  that  he  took  the 
mean  of  the  above  results,  4.713,  as  the  best  determination  of 
the  mean  density  of  the  earth. 

Hutton  prefers  to  take  2.77  as  the  mean  of  Playfair's  deter- 
minations for  the  density  of  the  hill,  and  the  density  of  the 
earth  as  f  of  2.77,  or  5  nearly.  In  a  paper  published  in  1821 
(52,  53  and  54),  Hutton  complains  that  his  share  in  the  Sche- 
hallien experiment  has  always  been  underestimated  ;  he  gives 
a  brief  account  of  the  observations,  calculations  and  results, 
and  considers  5  as  the  most  probable  value  of  the  mean  density 
of  the  earth.  He  shews  that  the  Schehallien  experiment  could 
not  be  made  to  give  the  same  result  as  that  of  Cavendish,  5.48, 
unless  the  deflection  11".  6  be  diminished  to  about  10". 5  or 
10". 4,  which  is  manifestly  too  great  an  error  to  have  been 
committed  by  Maskelyne,  considering  the  accuracy  of  the 
observer  and  of  the  instruments,  and  the  large  number  of  ob- 
servations made.  Hutton  suggests  the  repetition  of  the  ex- 
periment at  one  of  the  pyramids  in  Egypt.  Some  years  later 
Peters  (80£)  made  a  calculation  of  the  attraction  of  the  Great 
Pyramid. 

For  brief  accounts  of  the  Schehallien  experiment  and  criti- 
cisms upon  it,  reference  should  be  made  to  Hutton  (38  and  47, 
vol.  2,  pp.  69-77),  von  Zach  (43,  44  and  49),  Muncke  (61,  vol. 
3,  pp.  944-70),  Schmidt  (64,  vol.  2,  pp.  474-9),  Meuabrea  (71), 

55 


MEMOIRS    ON    THE    LAWS    OF    GRAVITATION 

Schell  (135),  Todhunter  (140,  vol.  1,  pp.  459-69).  Zanotti- 
Bianco  (148£,  pt.  2,  pp.  125-35)  and  Fresdorf  (186£,  pp.  5-7). 

Capt.  Jacob  has  remarked  (118  and  121)  that  by  this  method 
we  may  measure  the  attraction  of  the  mass  of  the  mountain 
above  the  surface,,  yet  we  do  not  know  how  much  ought  to  be 
added  or  subtracted  due  to  that  below  it. 

Von  Zach  makes  mention  of  several  early  astronomers  who 
assign  anomalies  in  their  geodetic  measurements  to  the  influence 
of  mountains  on  the  plumb-lines  of  their  instruments;  the 
reader  is  referred  to  von  Zach,  Humboldt  (82,  vol.  1,  notes, 
pp.  45-7)  and  Helmert  (148,  vol.  2,  chap.  4),  and  to  the  ac- 
count in  this  volume  (p.  123)  of  the  work  of  James  and 
Clarke.  Von  Zach  himself  made  a  very  careful  determination 
in  1810,  after  the  method  used  by  Bouguer,  of  the  attraction 
of  mount  Mimet,  near  Marseilles.  He  found  a  deflection  of 
the  plumb-line  amounting  to  2".  He  did  not  calculate  the 
density  of  the  earth.  His  observations  were  published  in  book 
form  in  1814  (49). 

For  this  work  Maskelyne  was  presented  by  the  Royal  Society 
with  the  Copley  medal.  At  the  presentation  the  President,  Sir 
John  Pringle,  delivered  an  address  (35)  on  the  attraction  of 
gravitation,  giving  a  critical  account  of  the  state  of  the  subject 
before  the  time  of  Newton,  as  well  as  of  its  later  developments. 

56 


EXPERIMENTS    TO    DETERMINE    THE 
DENSITY  OF  THE  EARTH 

BY 

HENRY   CAVENDISH,  ESQ.,   F.R.S.   AND  A.S. 
Read  June  21,  1798 


(From  the  Philosophical  Transactions  of  the  Royal  Society  of  London  for  tlie 
year  1798,  Part  II. ,  pp.  469-526) 


57 


CONTENTS 

PAGK 

Introduction 59 

Description  of  the  apparatus 61 

Method  of  observing  the  deflection 64 

"       "        "          "    time  of  vibration. 64 

Effect  of  the  resistance  of  the  air 65 

Account  of  the  experiments 67 

Testing  for  magnetic  effects 68 

Testing  the  elastic  properties  of  the  wire. 72 

Further  tests  for  magnetic  effects - 75 

Testing  tJie  effect  of  variation  of  temperature  about  the  box 76 

Final  observations 80 

On  the  tlieory  of  the  experiment 88 

Corrections  to  be  made  in  the  theory  as  first  given 91 

Effect  of  the  variable  position  of  the  arm  on  the  equations 97 

When  and  how  to  apply  the  corrections 98 

Table  of  results 99 

Conclusion 99 

Appendix  :  to  find  the  attraction  of  the  mahogany  case  on  the  balls 102 

58 


EXPERIMENTS    TO    DETERMINE    THE 
DENSITY  OF  THE  EARTH 

BY 

HENRY   CAVENDISH,   ESQ.,   F.R.S.   AND   A.S. 


MANY  years  ago,  the  late  Rev.  John  Miohell,  of  this  society, 
contrived  a  method  of  determining  the  density  of  the  earth,  hy 
rendering  sensible  the  attraction  of  small  quantities  of  matter  ; 
but,  as  he  was  engaged  in  other  pursuits,  he  did  not  complete 
the  apparatus  till  a  short  time  before  his  death,  and  did  not 
live  to  make  any  experiments  with  it.  After  his  death,  the 
apparatus  came  to  the  Rev.  Francis  John  Hyde  Wollaston, 
Jacksonian  Professor  at  Cambridge,  who,  not  having  conven- 
iences for  making  experiments  with  it,  in  the  manner  he  could 
wish,  was  so  good  as  to  give  it  to  me. 

The  apparatus  is  very  simple  ;  it  consists  of  a  wooden  arm,  6 
feet  long,  made  so  as  to  unite  great  strength  with  little  weight. 
This  arm  is  suspended  in  an  horizontal  position,  by  a  slender 
wire  40  inches  long,  and  to  each  extremity  is  hung  a  leaden 
ball,  about  2  inches  in  diameter  ;  and  the  whole  is  inclosed  in 
a  narrow  wooden  case,  to  defend  it  from  the  wind. 

As  no  more  force  is  required  to  make  this  arm  turn  round  on 
its  centre,  than  what  is  necessary  to  twist  the  suspending  wire, 
it  is  plain,  that  if  the  wire  is  sufficiently  slender,  the  most  min- 
ute force,  such  as  the  attraction  of  a  leaden  weight  a  few  inches 
in  diameter,  will  be  sufficient  to  draw  the  arm  sensibly  aside. 
The  weights  which  Mr.  Michell  intended  to  use  were  8  inches 
diameter.  One  of  these  was  to  be  placed  on  one  side  the  case, 
opposite  to  one  of  the  balls,  and  as  near  it  as  could  conveniently 
be  done,  and  the  other  on  the  other  side,  opposite  to  the  other 

59 


MEMOIRS    ON 

ball,  so  that  the  attraction  of  both  these  weights  would  con- 
spire in  drawing  the  arm  aside  ;  and,  when  its  position,  as  af- 
fected by  these  weights,  was  ascertained,  the  weights  were  to 
be  removed  to  the  other  side  of  the  case,  so  as  to  draw  the  arm 
the  contrary  way,  and  the  position  of  the  arm  was  to  be  again 
determined ;  and,  consequently,  half  the  difference  of  these  po- 
sitions would  shew  how  much  the  arm  was  drawn  aside  by  the 
attraction  of  the  weights. 

In  order  to  determine  from  hence  the  density  of  the  earth, 
it  is  necessary  to  ascertain  what  force  is  required  to  draw  the 
arm  aside  through  a  given  space.  This  Mr.  Michell  intended  to 
do,  by  putting  the  arm  in  motion,  and  observing  the  time  of  its 
vibrations,  from  which  it  may  easily  be  computed.* 

Mr.  Michell  had  prepared  two  wooden  stands,  on  which  the 
leaden  weights  were  to  be  supported,  and  pushed  forwards,  till 
they  came  almost  in  contact  with  the  case ;  but  he  seems  to 
have  intended  to  move  them  by  hand. 

As  the  force—with  which  the  balls  are  attracted  by  these 
weiglits  is  excessively  minute,  not  more  than  50)UOVuuo  °^ 
their  weight,  it  is  plain,  that  a  very  minute  disturbing  force 
will  be  sufficient  to  destroy  the  success  of  the  experiment;  and, 
from  the  following  experiments  it  will  appear,  that  the  disturb- 
ing force  most  difficult  to  guard  against,  is  that  arising  from 
the  variations  of  heat  and  cold  ;  for,  if  one  side  of  the  case  is 
warmer  than  the  other,  the  air  in  contact  with  it  will  be  rare- 
fied, and,  in  consequence,  will  ascend,  while  that  on  the  other 
side  will  descend,  and  produce  a  current  which  will  draw  the 
arm  sensibly  aside. f 

*  Mr.  Coulomb  has,  in  a  variety  of  cases,  used  a  contrivance  of  this  kind 
for  trying  small  attractions  ;  but  Mr.  Michell  informed  me  of  his  intention 
of  making  this  experiment,  and  of  the  method  he  intended  to  use,  before 
the  publication  of  any  of  Mr.  Coulomb's  experiments. 

f  M.  Cassini,  in  observing  the  variation  compass  placed  by  him  in  the 
observatory  (which  was  constructed  so  as  to  make  very  minute  changes  of 
position  visible,  nnd  in  which  the  needle  was  suspended  by  a  silk  thread), 
found  that  standing  near  the  box,  in  order  to  observe,  drew  the  needle 
sensibly  aside  ;  which  I  have  no  doubt  was  caused  by  this  current  of  air 
It  must  be  observed,  that  his  compass  box  was  of  metal,  which  transmits 
heat  faster  than  wood,  and  also  was  many  i nches  deep  ;  both  which  causes 
served  to  increase  the  current  of  air.  To  diminish  the  effect  of  this  cur- 
rent, it  is  by  all  means  advisable  to  make  the  box,  in  which  the  needle 
plays,  not  much  deeper  than  is  necessnry  to  prevent  the  needle  from  strik- 
ing against  the  top  and  bottom. 

60 


THE    LAWS    OF    GRAVITATION 

As  I  was  convinced  of  the  necessity  of  guarding  against  this 
source  of  error,  I  resolved  to  place  the  apparatus  in  a  room 
which  should  remain  constantly  shut,  and  to  ohserve  the  mo- 
tion of  the  arm  from  without,  hy  means  of  a  telescope  ;  and  to 
suspend  the  leaden  weights  in  such  manner,  that  I  could  move 
them  without  entering  into  the  room.  This  difference  in  the 
manner  of  observing,  rendered  it  necessary  to  make  some  al- 
teration in  Mr.  MichelFs  apparatus  ;  and,  as  there  were  some 
parts  of  it  which  I  thought  not  so  convenient  as  could  be 
wished,  I  chose  to  make  the  greatest  part  of  it  afresh. 

Fig.  1  is  a  longitudinal  vertical  section  through  the  instru- 
ment, and  the  building  in  which  it  is  placed.  ABCDDCB- 
AEFFE  is  the  case  ;  x  and  x  are  the  two  balls,  which  are  sus- 
pended by  the  wires  lix  from  the  arm  ghmh,  which  is  itself 
suspended  by  the  slender  wire  gl.  This  arm  consists  of  a 
slender  deal  rod  hnih,  strengthened  by  a  silver  wire  hyh',  by 
which  means  it  is  made  strong  enough  to  support  the*  balls, 
though  very  light.* 

The  case  is  supported,  and  set  horizontal,  by  four  screws, 
resting  on  posts  fixed  firmly  into  the  ground  ;  two  of  them  are 
represented  in  the  figure,  by  S  and  S ;  the  two  others  are  not 
represented,  to  avoid  confusion.  GG  and  GG  are  the  end 
walls  of  the  building.  W  and  W  are  the  leaden  weights  ; 
which  are  suspended  by  the  copper  rods  RrPrR,  and  the 
wooden  bar  rr,  from  the  centre  pin  Pp.  This  pin  passes 
through  a  hole  in  the  beam  HH,  perpendicularly  over  the  cen- 
tre of  the  instrument,  and  turns  round  in  it,  being  prevented 
from  falling  by  the  plate  p.  MM  is  a  pulley,  fastened  to  this 
pin;  and  Mm,  a  cord  wound  round  the  pulley,  and  passing 
through  the  end  wall ;  by  which  the  observer  may  turn  it 
round,  and  thereby  move  the  weights  from  one  situation  to  the 
other. 

Fig.  2  is  a  plan  of  the  instrument.  AAAA  is  the  case.  SSSS, 
the  four  screws  for  supporting  it.  hh,  the  arm  and  balls.  W 
and  W,  the  weights.  MM,  the  pulley  for  moving  them.  When 

*Mr.  Michell's  rod  was  entirely  of  wood,  and  was  much  stronger  and 
stiffer  than  this,  though  not  much  heavier;  but,  as  it  had  warped  when  it 
came  to  me,  I  chose  to  make  another,  and  preferred  this  form,  partly  as 
being  easier  to  construct  and  meeting  with  less  resistance  from  the  air, 
and  partly  because,  from  its  being  of  a  less  complicated  form,  I  could  more 
easily  compute  how  much  it  was  attracted  by  the  weights. 

61 


MEMOIRS    ON 


THE    LAWS    OF    GRAVITATION 

r' 

the  weights  are  in  this  position,  both  conspire  in  drawing  the 
arm  in  the  direction  7AV;  but,  when  they  are  removed  to  the 
situation  w  and  w,  represented  by  the  dotted  lines,  both  con- 
spire in  drawing  the  arm  in  the  contrary  direction  hw.  These 
weights  are  prevented  from  striking  the  instrument,  by  pieces 
of  wood,  which  stop  them  as  soon  as  they  come  within  £  of  an 
inch  of  the  case.  The  pieces  of  wood  are  fastened  to  the  wall 


of  the  building ;  and  I  find,  that  the  weights  may  strike  against 
them  with  considerable  force,  without  sensibly  shaking  the  in- 
strument. 

In  order  to  determine  the  situation  of  the  arm,  slips  of  ivory 
are  placed  within  the  case,  as  near  to  each  end  of  the  arm  as 
can  be  done  without  danger  of  touching  it,  and  are  divided  to 
20ths  of  an  inch.  Another  small  slip  of  ivory  is  placed  at  each 
and  of  the  arm,  serving  as  a  vernier,  and  subdividing  these 
divisions  into  5  parts  ;  so  that  the  position  of  the  arm  may  be 
observed  with  ease  to  lOOths  of  an  inch,  and  may  be  estimated 
to  less.  These  divisions  are  viewed,  by  means  of  the  short 
telescopes  T  and  T  (Fig.  1),  through  slits  cut  in  the  end  of  the 
case,  and  stopped  with  glass ;  they  are  enlightened  by  the  lamps 
L  and  L,  with  convex  glasses,  placed  so  as  to  throw  the  light 
on  the  divisions  ;  no  other  light  being  admitted  into  the  room. 

The  divisions  on  the  slips  of  ivory  run  in  the  direction  W«0 
(Fig.  2),  so  that,  when  the  weights  are  placed  in  the  positions 
w  and  w,  represented  by  the  dotted  circles,  the  arm  is  drawn 
aside,  in  such  direction  as  to  make  the  index  point  to  a  higher 
number  on  the  slips  of  ivory  ;  for  which  reason,  I  call  this  the 
positive  position  of  the  weights. 

FK  (Fig.  1)  is  a  wooden  rod,  which,  by  means  of  an  endless 
screw,  turns  round  the  support  to  which  the  wire  gl  is  fastened, 


MEMOIRS    ON 

and  thereby  enables  the  observer  to  turn  round  the  wire,  till 
the  arm  settles  in  the  middle  of  the  case,  without  danger  of 
touching  either  side.  The  wire  gl  is  fastened  to  its  support  at 
top,  and  to  the  centre  of  the  arm  at  bottom,  by  brass  clips,  in 
which  it  is  pinched  by  screws. 

In  these  two  figures,  the  different  parts  are  drawn  nearly  in 
the  proper  proportion  to  each  other,  and  on  a  scale  of  one  to 
thirteen. 

Before  I  proceed  to  the  account  of  the  experiments,  it  will 
be  proper  to  say  something  of  the  manner  of  observing.  Sup- 
pose the  arm  to  be  at  rest,  and  its  position  to  be  observed,  let 
the  weights  be  then  moved,  the  arm  will  not  only  be  drawn 
aside  thereby,  but  it  will  be  made  to  vibrate,  and  its  vibrations 
will  continue  a  great  while  ;  so  that,  in  order  to  determine  how 
much  the  arm  is  drawn  aside,  it  is  necessary  to  observe  the  ex- 
treme points  of  the  vibrations,  and  from  thence  to  determine 
the  point  which  it  would  rest  at  if  its  motion  was  destroyed,  or 
the  point  of  rest,  as  I  shall  call  it.  To  do  thia,  I  observe  three 
successive  extreme  points  of  a  vibration,  and  take  the  mean 
between  the  first  and  third  of  these  points,  as  the  extreme 
point  of  vibration  in  one  direction,  and  then  assume  the  mean 
between  this  and  the  second  extreme,  as  the  point  of  rest ;  for, 
as  the  vibrations  are  continually  diminishing,  it  is  evident,  that 
the  mean  between  two  extreme  points  will  not  give  the  true 
point  of  rest. 

It  may  be  thought  more  exact,  to  observe  many  extreme 
points  of  vibration,  so  as  to  find  the  point  of  rest  by  different 
sets  of  three  extremes,  and  to  take  the  mean  result ;  but  it 
must  be  observed,  that  notwithstanding  the  pains  taken  to  pre- 
vent any  disturbing  force,  the  arm  will  seldom  remain  perfect- 
ly at  rest  for  an  hour  together;  for  which  reason,  it  is  best  to 
determine  the  point  of  rest,  from  observations  made  as  soon 
after  the  motion  of  the  weights  as  possible. 

The  next  thing  to  be  determined  is  the  time  of  vibration, 
which  I  find  in  this  manner:  I  observe  the  two  extreme  points 
of  a  vibration,  and  also  the  times  at  which  the  arm  arrives  at  two 
given  divisions  between  these  extremes,  taking  care,  as  well  as  I 
can  guess,  that  these  divisions  shall  be  on  different  sides  of  the 
middle  point,  and  not  very  far  from  it.  I  then  compute  the 
middle  point  of  the  vibration,  and,  by  proportion,  find  the  time 
at  which  the  arm  comes  to  this  middle  point.  I  then,  after  a 

64 


THE    LAWS    OF    GRAVITATION 


number  of  vibrations,  repeat  this  operation,  and  divide  the  in- 
terval of  time,  between  the  coming  of  the  arm  to  these  two 
middle  points,  by  the  number  of  vibrations,  which  gives  the 
time  of  one  vibration.  The  following  example  will  explain 
what  is  here  said  more  clearly : 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle 
of  vibration 

27.2 

25 
24 

10h  23'     4"  | 
57     f 

— 

10h  23'  23" 

22.1 



246 

27. 





24.7 

22.6 





2475 

26.8 



— 

24.8 

23. 

— 

— 

24.85 

26.6 



— 

24.9 

25 
24 

11      5   22    ) 

6   48    J 

— 

11      5  22 

23.4 

The  first  column  contains  the  extreme  points  of  the  vibra- 
tions. The  second,  the  intermediate  divisions.  The  third, 
the  time  at  which  the  arm  came  to  these  divisions ;  and  the 
fourth,  the  point  of  rest,  which  is  thus  found  :  the  mean  be- 
tween the  first  and  third  extreme  points  is  27.1,  and  the  mean 
between  this  and  the  second  extreme  point  is  24.6,  which  is 
the  point  of  rest,  as  found  by  the  three  first  extremes.  In  like 
manner,  the  point  of  rest  found  by  the  second,  third,  and 
fourth  extremes,  is  24.7,  and  so  on.  The  fifth  column  is  the 
time  at  which  the  arm  came  to  the  middle  point  of  the  vibra- 
tion, which  is  thus  found:  the  mean  between  27.2  and  22.1  is 
24.65,  and  is  the  middle  point  of  the  first  vibration  ;  and,  as 
the  arm  came  to  25  at  10h  23'  4",  and  to  24  at  10h  23'  57",  we 
find,  by  proportion,  that  it  came  to  24.65  at  10h  23'  23".  In 
like  manner,  the  arm  came  to  the  middle  of  the  seventh  vibra- 
tion at  llh  5'  22";  and,  therefore,  six  vibrations  were  performed 
in  41'  59",  or  one  vibration  in  7'  0". 

To  judge  of  the  propriety  of  this  method,  we  must  consider 
in  what  manner  the  vibration  is  affected  by  the  resistance  of  the 
airland  by  the  motion  of  the  point  of  rest. 
f/Let  the  arm,  during  the  first  vibration,  move  from  D  to  B 
(Fig.  3),  and,  during  the  second,  from  B  to  d ;  Br7  being  less 
than  DB,  on  account  of  the  resistance.  Bisect  DB  in  M,  and 
E  65 


MEMOIRS    ON 

Bi7  in  m,  and  bisect  Mm  in  n,  and  let  x  be  any  point  in  the 
vibration  ;  then,  if  the  resistance  is  proportional  to  the  square 
of  the  velocity,  the  whole  time  of  a  vibration  is  very  little  al- 
tered ;  but,  if  T  is  taken  to  the  time  of  one  vibration,  as  the 
diameter  of  a  circle  to  its  semi  -circumference,  the  time  of 

TxDd 
moving  from  B  to  n  exceeds  -J  a  vibration,  by      ^^        nearly  ; 

and  the  time  of  moving  from  B  to  m  falls  short  of  J  a  vibration, 

D      6  M  x  6 

1  -  >—  3  -  h~t-t  -  j  -  -* 

Fig.  3 

by  as  much  ;  and  the  time  of  moving  from  B  to  x,  in  the  sec- 
ond vibration,  exceeds  that  of  moving  from  x  to  B,  in  the  first, 


.  . 

by  -  _  -  ,  supposing  Da  to  be  bisected  m  3  ;  so  that, 


if  a  mean  is  taken,  between  the  time  of  the  first  arrival  of 
the  arm  at  x  and  its  returning  back  to  the  same  point,  this 
mean  will  be  earlier*  than  the  true  time  of  its  coming  to  B,  by 


The  effect  of  motion  in  the  point  of  rest  is,  that  when  the 
arm  is  moving  in  the  same  direction  as  the  point  of  rest,  the 
time  of  moving  from  one  extreme  point  of  vibration  to  the 
other  is  increased,  and  it  is  diminished  when  they  are  moving 
in  contrary  directions  ;  but,  if  the  point  of  rest  moves  uni- 
formly, the  time  of  moving  from  one  extreme  to  the  middle 
point  of  the  vibration,  will  be  equal  to  that  of  moving  from 
the  middle  point  to  the  other  extreme,  and,  moreover,  the  time 
of  two  successive  vibrations  will  be  very  little  altered  ;  and, 
therefore,  the  time  of  moving  from  the  middle  point  of  one 
vibration  to  the  middle  point  of  the  next,  will  also  be  very 
little  altered. 

*  [This  word  should  be  "later"  as  is  observed  by  Todhunter  (140,  vol.  2,  p. 
165).  For  an  elementary  discussion  of  this  kind  of  motion  see  Williamson  and 
Tarleton's  "  Treatise  of  Dynamics,"  ex.  13,  §  117.  Poisson  (65,  vol.  1,  pp. 
353-361)  and  Menabrea  (71)  have  given  very  elaborate  analyses  of  the  problem. 
Cornu  and  Bailie  (137,  141,  142,  143,  and  157)  proved  in  1878  that  the  re- 
sistance in  the  case  under  consideration  is  proportional  to  the  first  power  of 
the  velocity.] 

66 


THE    LAWS    OF    GRAVITATION 

It  appears,  therefore,  that  on  account  of  the  resistance  of 
the  air,  the  time  at  which  the  arm  comes  to  the  middle  point 
of  the  vibration,  is  not  exactly  the  mean  between  the  times  of 
its  coming  to  the  extreme  points,  which  causes  some  inaccur- 
acy in  my  method  of  finding  the  time  of  a  vibration.  It  must 
be  observed,  however,  that  as  the  time  of  coming  to  the  middle 
point  is  before  the  middle  of  the  vibration,  both  in  the  first 
and  last  vibration,  and  in  general  is  nearly  equally  so,  the  error 
produced  from  this  cause  must  be  inconsiderable ;  and,  on  the 
whole,  I  see  no  method  of  finding  the  time  of  a  vibration  which 
is  liable  to  less  objection. 

The  time  of  a  vibration  may  be  determined,  either  by  previous 
trials,  or  it  may  be  done  at  each  experiment,  by  ascertaining  the 
time  of  the  vibrations  which  the  arm  is  actually  put  into  by 
the  motion  of  the  weights ;  but  there  is  one  advantage  in  the 
latter  method,  namely,  that  if  there  should  be  any  accidental 
attraction,  such  as  electricity,  in  the  glass  plates  through  which 
the  motion  of  the  arm  is  seen,  which  should  increase  the  force 
necessary  to  draw  the  arm  aside,  it  would  also  diminish  the 
time  of  vibration  ;  and,  consequently,  the  error  in  the  result 
would  be  much  less,  when  the  force  required  to  draw  the  arm 
aside  was  deduced  from  experiments  made  at  the  time,  than 
when  it  was  taken  from  previous  experiments. 

ACCOUNT  OF  THE  EXPERIMENTS 

In  my  first  experiments,  the  wire  by  which  the  arm  was  sus- 
pended was  39£  inches  long,  and  was  of  copper  silvered,  one 
foot  of  which  weighed  2T4^  grains ;  its  stiffness  was  such  as  to 
make  the  arm  perform  a  vibration  in  about  15  minutes.  I  im- 
mediately found,  indeed,  that  it  was  not  stiff  enough,  as  the 
attraction  of  the  weights  drew  the  balls  so  much  aside,  as  to 
make  them  touch  the  sides  of  the  case ;  I,  however,  chose  to 
make  some  experiments  with  it,  before  I  changed  it. 

In  this  trial,  the  rods  by  which  the  leaden  weights  were  sus- 
pended were  of  iron ;  for,  as  I  had  taken  care  that  there  should 
be  nothing  magnetical  in  the  arm,  it  seemed  of  no  signification 
whether  the  rods  were  magnetical  or  not ;  but,  for  greater  se- 
curity, I  took  off  the  leaden  weights,  and  tried  what  effect  the 
rods  would  have  by  themselves.  Now  I  find,  by  computation, 
that  the  attraction  of  gravity  of  these  rods  on  the  balls,  is  to 

67 


MEMOIRS    ON 

that  of  the  weights,  nearly  as  17  to  2500  ;  so  that,  as  the  at- 
traction of  the  weights  appeared,  by  the  foregoing  trial,  to  be 
sufficient  to  draw  the  arm  aside  by  about  15  divisions,  the  at- 
traction of  the  rods  alone  should  draw  it  aside  about  -fa  of  a 
division  ;  and,  therefore,  the  motion  of  the  rods  from  one  near 
position  to  the  other,  should  move  it  about  ^  of  a  division. 

The  result  of  the  experiment  was,  that  for  the  first  15  min- 
utes after  the  rods  were  removed  from  one  near  position  to  the 
other,  very  little  motion  was  produced  in  the  arm,  and  hardly 
more  than  ought  to  be  produced  by  the  action  of  gravity  ;  but 
the  motion  then  increased,  so  that,  in  about  a  quarter  or  half 
an  hour  more,  it  was  found  to  have  moved  }  or  1J  division,  in 
the  same  direction  that  it  ought  to  have  done  by  the  action  of 
gravity.  On  returning  the  irons  back  to  their  former  position, 
the  arm  moved  backward,  in  the  same  manner  that  it  before 
moved  forward. 

It  must  be  observed,  that  the  motion  of  the  arm,  in  these  ex- 
periments, was  hardly  more  than  would  sometimes  take  place 
without  any  apparent  cause;  but  yet,  as  in  three  experiments 
which  were  made  with  these  rods,  the  motion  was  constantly  of 
the  same  kind,  though  differing  in  quantity  from  J  to  1£  divis- 
ion, there  seems  great  reason  to  think  that  it  was  produced  by 
the  rods. 

As  this  effect  seemed  to  me  to  be  owing  to  magnetism,  though 
it  was  not  such  as  I  should  have  expected  from  that  cause,  I 
changed  the  iron  rods  for  copper,  and  tried  them  as  before  ; 
the  result  was,  that  there  still  seemed  to  be  some  effect  of  the 
same  kind,  but  more  irregular,  so  that  I  attributed  it  to  some 
accidental  cause,  and  therefore  hung  on  the  leaden  weights,  and 
proceeded  with  the  experiments. 

It  must  be  observed,  that  the  effect  which  seemed  to  be  pro- 
duced by  moving  the  iron  rods  from  one  near  position  to  the 
other,  was,  at  a  medium,  not  more  than  one  division  ;  whereas 
the  effect  produced  by  moving  the  weight  from  the  midway  to 
the  near  position,  was  about  15  divisions  ;  so  that,  if  I  had  con- 
tinued to  use  the  iron  rods,  the  error  in  the  result  caused  there- 
by, could  hardly  have  exceeded  -^  of  the  whole. 

68 


THE    LAWS    OF    GRAVITATION 


EXPERIMENT  I.     AUG.  5 
Weights  in  midway  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  mid.  of 
vibration 

Difference 

11.4 
11.5 
11.5 

9h  42'     0" 
55     0 
10      5     0 

11.5 

23.4 
27.6 
24.7 
27,3 
25.1 


At  10h  5',  weights  moved  to  positive  position 


25.82 
26.07 
26.1 


At  llh  6',  weights  returned  back  to  midway  position 


5. 

11 
12 

0   0  48  } 
1  30  f 

— 

oh  r  13" 

18.2 

12 

\  — 

14'  56" 

12 
11 

16  29  ) 
17  20  <i 

— 

16  9 

6.6 

11.92 

— 

14  36 

11 
12 

30  24  ) 
31  11  f 

— 

30  45 

16.3 

11.72 

— 

15  13 

12 
11 

45  58  \ 

47  4  f 

— 

45  58 

7.7 

Motion  on  moving  from  midway  to  pos.  =14.32 

pos.  to  midway  =  14.1 
Time  of  one  vibration  =  14'  55" 

It  must  be  observed,  that  in  this  experiment,  the  attraction 
of  the  weights  drew  the  arm  from  11.5  to  25.8,  so  that,  if  no 
contrivance  had  been  used  to  prevent  it,  the  momentum  ac- 
quired thereby  would  have  carried  it  to  near  40,  and  would, 
therefore,  have  made  the  balls  to  strike  against  the  case.  To 
prevent  this,  after  the  arm  had  moved  near  15  divisions,  I  re- 
turned the  weights  to  the  midway  position,  and  let  them  re- 
main there,  till  the  arm  came  nearly  to  the  extent  of  its  vibra- 
tion, and  then  again  moved  them  to  the  positive  position, 
whereby  the  vibrations  were  so  much  diminished,  that  the  balls 
did  not  touch  the  sides ;  and  it  was  this  which  prevented  my 
observing  the  first  extremity  of  the  vibration.  A  like  method 
was  used,  when  the  weights  were  returned  to  the  midway  posi- 
tion, and  in  the  two  following  experiments. 

69 


MEMOIRS    ON 


The  vibrations,  in  moving  the  weights  from  the  midway  to 
the  positive  position,  were  so  small,  that  it  was  thought  not 
worth  while  to  observe  the  time  of  the  vibration.  When  the 
weights  were  returned  to  the  midway  position,  I  determined 
the  time  of  the  arm's  coming  to  the  middle  point  of  each  vibra- 
tion, in  order  to  see  how  nearly  the  times  of  the  different 
vibrations  agreed  together.  In  great  part  of  the  following  ex- 
periments, 1  contented  myself  with  observing  the  time  of  its 
coming  to  the  middle  point  of  only  the  first  and  last  vibration. 

EXPERIMENT  II.     AUG.  6 

Weights  in  midway  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  mid.  of 
vibration 

Difference 

11 

10h     4'      0" 

11 

11       0 

11 

17      0 

11 

25     0 

11. 

Weights  moved  to  positive  position 


29.3 

24.1 

30. 

26.2 

29.7 

26.1) 

28-7 

27.1 

28.4 


26.87 

27.57 
28.02 
28.12 
28.05 
27.85 
27.82 


Weights  returned  to  midway  position 


6. 

12 

1   3 

50  ) 

13 

4 

34  \ 

18.5 

— 

_^_ 

13 

18 

29  ) 

12 

19 

18  \ 

6.5 

— 

— 

11 

33 

48  ) 

12 

34 

51  \ 

15.2 





13 

45 

8  { 

12 

46 

22  J 

7.1 





11 

2   3 

48  ) 

12 

5 

18  f 

13.6 

12.37 


11.67 


11. 


10.75 


lh 

4' 

1" 

18 

53 

33 

39 

47 

25 

2 

2 

50 

14  46 


13  46 


15  25 


Motion  of  arm  on  moving  weights  from  midway  to  pos.  =  15.87 

pos.  to  midway  =  15.45 

Time  of  one  vibration =14'  42" 

70 


THE    LAWS    OF    GRAVITATION 


EXPERIMENT  III.     AUG.  7 

The  weights  being  in  the  positive  position,  and  the  arm  a  little  in  motion 


Exiiviue 
IMHIIIS 

Divisions 

Time 

Point  of 
rest 

Time  of  mid.  of 
vibration 

Difference 

31.5 

29 



— 

30.12 

31 





30.02 

29.1 

Weights  moved  to  midway  position 

10h  34'  55" 

49  39 

11      4  17 
19     4 

33  31 
Mon 

0     2  59 

47  40 


Motion  of  the  arm  on  moving  weights  from  pos.  to  mid.=  15.22 

mid.  to  pos.=:  14.5 

Time  of  one  vibration,  when  in  mid.  position  =  14'  39" 

pos.  position  =  14'  54" 


9. 

14 

10h  34'  18"  | 

15 

35     8    f 

20.5 



— 

14.8 

15 

49   31    \ 

14 

50   27    f 

9.2 



— 

14.07 

14 

11      5     7    ) 

15 

6    18    f 

17.4 





13.52 

14 

18   46    } 

13 

19   58    f 

10.1 



— 

13.3 

13 

33   46    ) 

14 

35  26    f 

15.6 

Weights  moved  to  positive  p< 

32. 

28 

0      2   48    ) 

27 

3   56    f 

23,7 

— 

— 

27.8 

31.8 





28.27 

25.8 

— 



28.62 

27 

44  58    | 

28 

46   50    f 

31.1 

14'  44" 
14  38 
1447 
1427 


•  These  experiments  are  sufficient  to  shew,  that  the  attraction 
of  the  weights  on  the  balls  is  very  sensible,  and  are  also  suf- 
ficiently regular  to  determine  the  quantity  of  this  attraction 
pretty  nearly,  as  the  extreme  results  do  not  differ  from  each 
other  by  more  than  -^  part.  But  there  is  a  circumstance  in 
them,  the  reason  of  which  does  not  readily  appear,  namely, 
that  the  effect  of  the  attraction  seems  to  increase,  for  half  an 

71 


MEMOIRS    ON 

hour,  or  an  hour,  after  the  motion  of  the  weights ;  as  it  may 
be  observed,  that  in  all  three  experiments,  the  mean  position 
kept  increasing  for  that  time,  after  moving  the  weights  to  the 
positive  position  ;  and  kept  decreasing,  after  moving  them 
from  the  positive  to  the  midway  position. 

The  first  cause  which  occurred  to  me  was,  that  possibly  there 
might  be  a  want  of  elasticity,  either  in  the  suspending  wire,  or 
something  it  was  fastened  to,  which  might  make  it  yield  more 
to  a  given  pressure,  after  a  long  continuance  of  that  pressure, 
than  it  did  at  first. 

To  put  this  to  the  trial,  I  moved  the  index  so  much,  that  the 
arm,  if  not  prevented  by  the  sides  of  the  case,  would  have  stood 
at  about  50  divisions,  so  that,  as  it  could  not  move  farther  than 
to  35  divisions,  it  was  kept  in  a  position  15  divisions  distant 
from  that  which  it  would  naturally  have  assumed  from  the 
stiffness  of  the  wire  ;  or,  in  other  words,  the  wire  was  twisted 
15  divisions.  After  having  remained  two  or  three  hours  in  this 
position,  the  index  was  moved  back,  so  as  to  leave  the  arm  at 
liberty  to  assume  its  natural  position. 

It  must  be  observed,  that  if  a  wire  is  twisted  only  a  little 
more  than  its  elasticity  admits  of,  then,  instead  of  setting,  as 
it  is  called,  or  acquiring  a  permanent  twist  all  at  once,  it  sets 
gradually,  and,  when  it  is  left  at  liberty,  it  gradually  loses  part 
of  that  set  which  it  acquired  ;  so  that  if,  in  this  experiment, 
the  wire,  by  having  been  kept  twisted  for  two  or  three  hours, 
had  gradually  yielded  to  this  pressure,  or  had  begun  to  set,  it 
would  gradually  restore  itself,  when  left  at  liberty,  and  the 
point  of  rest  would  gradually  move  backwards  ;  but,  though 
the  experiment  was  twice  repeated,  I  could  not  perceive  any 
such  effect. 

The  arm  was  next  suspended  by  a  stiffer  wire. 


EXPERIMENT  IV.     AUG.  12 
Weights  in  midway  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  mid.  of 
vibration 

Difference 

21.6 

9h  30'     0" 

21.5 

52     0 

21.5 

10    13     0 

21.5 

72 


THE    LAWS    OF    GRAVITATION 

Weights  moved  from  midway  to  positive  position 


27.2 

22.1 



— 

24.6 

27. 



— 

24.67 

22.6 

— 

— 

24.75 

26.8 



— 

248 

23.0 

— 

— 

24.85 

26.6 

— 

— 

24.9 

23.4 

Weights  moved  to  negative  position 

15. 

17 
19 

19   25    ) 
20  41    j 

— 

'  10h  20'  31'' 

22.4 

— 

— 

18.72 

— 

7'   0" 

20 
19 

26  45    I 

27  22    \ 

— 

27    31 

15.1 

— 

— 

18.52 

— 

6  57 

19 
20 

35     1    ) 

48    f 

— 

34   28 

21.5 

— 

18.35 

— 

7  23 

20 
19 

40  23    ) 

41  18    j" 

— 

41    51 

15.3 

— 

— 

18.22 

— 

6  48 

18 
19 

48  36    ) 
49  24    j" 

— 

48   39 

20.8 





18.1 



6  58 

19 

18 

54  45    ) 
55  45    J 

— 

55   37 

15.5 

Weights  moved  to  positive  position 

31.3 

25 
23 

11    10  25    I 
11     3    J 

— 

11    10   40 

17.1 



— 

24.02 



7    3 

22 
23 

17     6    ) 
26    f 

— 

17   43 

30.6 





24.17 



7    1 

25 
23 

24  33    ) 

25  17    f 

— 

24  44 

18.4 





2432 



7    5 

23 
25 

31  21    ) 
32     9    \ 

— 

31    49 

29.9 





24.4 



6  59 

25 
23 

38  39    } 
39  31     f 

— 

38   48 

19.4 

— 

— 

24.5 

— 

7    6 

23 
25 

45  16    / 
46   12    f 

— 

45  54 

29.3 

Moiion  of  arm  on  moving  weights  from  midway  to  pos.  =  3.1 

pos.  to  neg.  =  6.18 

neg.  to  pos.  —  5.92 

Time  of  one  vibration  in  neg.  position                                =1'  1" 

pos.  position                                =7'  3" 

73 

MEMOIRS    ON 

EXPERIMENT  V.     AUG.  20 

The  weights  being  in  the  positive  position,  the  arm  was  made  to  vibrate,  by 
moving  the  index 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  mid.  of 
vibration 

Difference 

29.6 
21.1 
29. 
21.6 

— 

— 

25.2 

25.17 

Weights  moved  to  negative  position 


22.6 

V 

20 
19 

10h  22'   47"  ) 
23    30    f 

— 

10h  23'    11" 

16.3 

— 

— 

19.27 

21.9 





19.15 

16.5 



— 

19.1 

21.5 

— 

— 

19.07 

16.8 

— 

— 

19.07 

21.2 





19.07 

17.1 

— 

— 

19.05 

20.8 

— 

— 

19.02 

17.4 





19.05 

20.6 

— 



19.02 

20 
19 

11     32    16    ) 

33    58    j 

— 

11     33    53 

17.5 

— 

— 

18.97 

— 

7'    13" 

19 
20 

41     16    ) 
43      0    f 

— 

41      6 

20.3 

Weights  moved  to  positive  position 

20.2 

24 
26 

49    10    / 
50     19    f 

— 

49    37 

29/4 

— 

— 

24.95 

— 

7    7 

26 
25 

56    15    ) 

47    f 

— 

\      56    44 

20.8 

—  . 

24  92 

28.7 





24.87 

21.3 

— 



24.85 

28.1 

— 

— 

24.75 

21.5 





24.67 

27.6 





24.67 

22. 

— 



24.7 

24 
25 

0    45    48    | 
46    43    f 

—  . 

0    46    21 

27.2 

— 



24.7 



7    1 

25 
24 

53    11    ) 
54      9    f 

— 

53    22 

22.4 

Motion  of  arm  on  moving  weights  from  pos.  to  neg.  =  5.9 

neg.  to  pos.  =  5.98 

Time  of  one  vibration,  when  weights  are  in  neg.  position  =  7'  5" 

pos.  position  =  7'  5" 

74 

THE    LAWS    OF    GRAVITATION 

In  the  fourth  experiment,  the  effect  of  the  weights  seemed 
to  increase  on  standing,  in  all  three  motions  of  the  weights, 
conformably  to  what  was  observed  with  the  former  wire  ;  but 
in  the  last  experiment  the  case  was  different ;  for  though,  on 
moving  the  weights  from  positive  to  negative,  the  effect  seemed 
to  increase  on  standing,  yet,  on  moving  them  from  negative  to 
positive,  it  diminished. 

My  next  trials  were,  to  see  whether  this  effect  was  owing  to 
magnetism.  Now,  as  it  happened,  the  case  in  which  the  arm 
was  inclosed,  was  placed  nearly  parallel  to  the  magnetic  east 
and  west,  and  therefore,  if  there  was  anything  magnetic  in  the 
balls  and  weights,  the  balls  would  acquire  polarity  from  the 
earth  ;  and  the  weights  also,  after  having  remained  some  time, 
either  in  the  positive  or  negative  position,  would  acquire  po- 
larity in  the  same  direction,  and  would  attract  the  balls;  but, 
when  the  weights  were  moved  to  the  contrary  position,  that 
pole  which  before  pointed  to  the  north,  would  point  to  the 
south,  and  would  repel  the  ball  it  was  approached  to  ;  but  yet, 
as  repelling  one  ball  towards  the  south  has  the  same  effect  on 
the  arm  as  attracting  the  other  towards  the  north,  this  would 
have  no  effect  on  the  position  of  the  arm.  After  some  time, 
however,  the  poles  of  the  weight  would  be  reversed,  and  would 
begin  to  attract  the  balls,  and  would  therefore  produce  the 
same  kind  of  effect  as  was  actually  observed. 

To  try  whether  this  was  the  case,  I  detached  the  weights 
from  the  upper  part  of  the  copper  rods  by  which  they  were 
suspended,  but  still  retained  the  lower  joint,  namely,  that 
which  passed  through  them  ;  I  then  fixed  them  in  their  posi- 
tive position,  in  such  manner,  that  they  could  turn  round  on 
this  joint,  as  a  vertical  axis.  I  also  made  an  apparatus  by 
which  I  could  turn  them  half  way  round,  on  these  vertical  axes, 
without  opening  the  door  of  the  room. 

Having  suffered  the  apparatus  to  remain  in  this  manner  for  a 
day,  I  next  morning  observed  the  arm,  and,  having  found  it  to 
be  stationary,  turned  the  weights  half  way  round  on  their  axes, 
but  could  not  perceive  any  motion  in  the  arm.  Having  suf- 
fered the  weights  to  remain  in  this  position  for  about  an  hour, 
I  turned  them  back  into  their  former  position,  but  without  its 
having  any  effect  on  the  arm.  This  experiment  was  repeated 
on  two  other  days,  with  the  same  result. 

We  may  be  sure,  therefore,  that  the  effect  in  question  could 

75 


MEMOIRS    ON 

not  be  produced  by  magnetism  in  the  weights  ;  for,  if  it  was, 
turning  them  half  round  on  their  axes,  would  immediately  have 
changed  their  magnetic  attraction  into  repulsion,  and  have 
produced  a  motion  in  the  arm. 

As  a  further  proof  of  this,  I  took  off  the  leaden  weights,  and 
in  their  room  placed  two  10-inch  magnets  ;  the  apparatus  for 
turning  them  round  being  left  as  it  was,  and  the  magnets  being 
placed  horizontal,  and  pointing  to  the  balls,  and  with  their 
north  poles  turned  to  the  north  ;  but  I  could  not  find  that 
any  alteration  was  produced  in  the  place  of  the  arm,  by  turn- 
ing them  half  round  ;  which  not  only  confirms  the  deduction 
drawn  from  the  former  experiment,  but  also  seems  to  shew,  that 
in  the  experiments  with  the  iron  rods,  the  effect  produced  could 
not  be  owing  to  magnetism. 

The  next  thing  which  suggested  itself  to  me  was,  that  pos- 
sibly the  effect  might  be  owing  to  a  difference  of  tempera- 
ture between  the  weights  and  the  case  ;  for  it  is  evident, 
that  if  the  weights  were  much  warmer  than  the  case,  they 
would  warm  that  side  which  was  next  to  them,  and  produce  a 
current  of  air,  which  would  make  the  balls  approach  nearer  to 
the  weights.  Though  I  thought  it  not  likely  that  there  should 
be  sufficient  difference,  between  the  heat  of  the  weights  and 
case,  to  have  any  sensible  effect,  and  though  it  seemed  im- 
probable that,  in  all  the  foregoing  experiments,  the  weights 
should  happen  to  be  warmer  than  the  case,  I  resolved  to  ex- 
amine into  it,  and  for  this  purpose  removed  the  apparatus  used 
in  the  last  experiments,  and  supported  the  weights  by  the  cop- 
per rods,  as  before  ;  and,  having  placed  them  in  the  midway 
position,  I  put  a  lamp  under  each,  and  placed  a  thermometer 
with  its  ball  close  to  the  outside  of  the  case,  near  that  part 
which  one  of  the  weights  approached  to  in  its  positive  position, 
and  in  such  manner  that  I  conld  distinguish  the  divisions  by 
the  telescope.  Having  done  this,  I  shut  the  door,  and  some 
time  after  moved  the  weights  to  the  positive  position.  At  first, 
the  arm  was  drawn  aside  only  in  its  usual  manner  ;  but,  in 
half  an  hour,  the  effect  was  so  much  increased,  that  the  arm 
was  drawn  14  divisions  aside,  instead  of  about  three,  as  it 
would  otherwise  have  been,  and  the  thermometer  was  raised 
near  1°.5  ;  namely,  from  61°  to  62°.5.  On  opening  the  door, 
the  weights  were  found  to  be  no  more  heated,  than  just  to  pre- 
vent their  feeling  cool  to  my  fingers. 

76 


THE    LAWS    OF    GRAVITATON 

As  the  effect  of  a  difference  of  temperature  appeared  to  be 
so  great,  I  bored  a  small  hole  in  one  of  the  weights,  about 
three-quarters  of  an  inch  deep,  and  inserted  the  ball  of  a  small 
thermometer,  and  then  covered  up  the  opening  with  cement. 
Another  small  thermometer  was  placed  with  its  ball  close  to 
the  case,  and  as  near  to  that  part  to  which  the  weight  was 
approached  as  could  be  done  with  safety  ;  the  thermometers 
being  so  placed,  that  when  the  weights  were  in  the  negative 
position,  both  could  be  seen  through  one  of  the  telescopes,  by 
means  of  light  reflected  from  a  concave  mirror. 


EXPERIMENT  VI.     SEPT.  6 

Weights  in  midway  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Thermometer 

in   air 

in  weight 

18.9 

18.85 

9h   43' 

10      3 

18.85 

55.5 

13.1 
18.4 
13.4 
missed 
13.6 
17.6 
13.8 
17.4 
14.0 
17.2 


25.8 
17.5 
25.4 
18.1 
25.0 
missed 
24.7 
19. 
24.4 


Weights  moved  to  negative  position 
10 


11 


12 



18 

15.82 

25 

39 

_ 

46 

15.65 

53 

15.65 

0 

15.65 

7 

15.65 

14 

— 

55.5 


55.5 


55.5 


55.8 


55.8 


Weights  moved  to  positive  position 


23 

30 

21 

.55 

37 

21 

.6 

44 

21 

05 

51 

0 

5 

12 

21 

.77 

19 

Motion  of  arm  on  moving  weights  from  midway  to  —  =  3.03 

-to +=5. 9 


77 


MEMOIRS    ON 


EXPERIMENT  VII.     SEPT.  18 

Weights  in  midway  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Thermometer 

in  air 

in  weight 

19.4 
19.4 

8h  30' 
9    32 

— 

56.7 
56.6 

Weights  moved  to  negative  position 


13.6 

18.8 
13.8 

- 

40 

47 
54 

16.25 

57.2 

16.9 
14.  J 
16.6 

— 

Eight  extrer 

10    58 
11      5 
12 

ne  points  miss 
15.62 

ed 

26.4 
17.2 
26.1 

' 

Veights  moved 
20 
28 
35 

to  positive  po& 
21.72 

ition 
56.5 

19.3 
25.1 
19.7 

— 

Four  extren 
0    10 
17 
24 

le  points  miss 
22.3 

ed 

Motion  of  arm  on  moving  weights  from  midway  to—  =  3.15 
-  to  +  =  6.1 

EXPERIMENT  VIII.     SEPT.  23 
in  midway  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Thermometer 

in  air 

in  weight 

19.3 
19.2 

9h  46' 
10    45 

19.2 

53.1 
53.1 

Weights  moved  to  negative  position 


13.5 

— 

56 





18.6 

— 

11      3 

16.07 

13.6 

— 

10 

Four  extreme  points  missed 

17.4 

— 

44 

14.1 

— 

51 

15.7 

17.2 

— 

58 

— 

— 

53.6 


f  O     *• 

53.6 


78 


15.7 
26.7 
16.6 


25.9 
18.1 
25.5 


THE    LAWS    OF    GRAVITATION 

Weights  moved  to  positive  position 
Oh     1' 


8 

15 


21.42 


53.15 


Two  extreme  points  missed 
36 


43 

50 


21.9 


Motion  of  arm  on  moving  weights  from  midway  to  —=3.13 

-to  +=5.  72 


In  these  three  experiments,  the  effect  of  the  weight  appeared 
to  increase  from  two  to  five  tenths  of  a  division,  on  standing 
an  hour  ;  and  the  thermometers  shewed,  that  the  weights  were 
three  or  five  tenths  of  a  degree  warmer  than  the  air  close  to  the 
case.  In  the  two  last  experiments,  I  put  a  lamp  into  the  room, 
over  night,  in  hopes  of  making  the  air  warmer  than  the  weights, 
but  without  effect,  as  the  heat  of  the  weights  exceeded  that  of 
the  air  more  in  these  two  experiments  than  in  the  former. 

On  the  evening  of  October  17,  the  weights  being  placed  in 
the  midway  position,  lamps  were  put  under  them,  in  order  to 
warm  them  ;  the  door  was  then  shut,  and  the  lamps  suffered  to 
burn  out.  The  next  morning  it  was  found,  on  moving  the 
weights  to  the  negative  position,  that  they  were  7°.  5  warmer 
than  the  air  near  the  case.  After  they  had  continued  an  hour 
in  that  position,  they  were  found  to  have  cooled  1°.5,  so  as  to 
be  only  6°  warmer  than  the  air.  They  were  then  moved  to  the 
positive  position  ;  and  in  both  positions  the  arm  was  drawn 
aside  about  four  divisions  more,  after  the  weights  had  remained 
an  hour  in  that  position,  than  it  was  at  first. 

May  22,  1798.  The  experiment  was  repeated  in  the  same 
manner,  except  that  the  lamps  were  made  so  as  to  burn  only  a 
short  time,  and  only  two  hours  were  suffered  to  elapse  before 
the  weights  were  moved.  The  weights  were  now  found  to  be 
scarcely  2°  warmer  than  the  case  ;  and  the  arm  was  drawn 
aside  about  two  divisions  more,  after  the  weights  had  remained 
an  hour  in  the  position  they  were  moved  to,  than  it  was  at  first. 

On  May  23,  the  experiment  was  tried  in  the  same  manner, 
except  that  the  weights  were  cooled  by  laying  ice  on  them;  the 
ice  being  confined  in  its  place  by  tin  plates,  which,  on  moving 
the  weights,  fell  to  the  ground,  so  as  not  to  be  in  the  way.  On 
moving  the  weights  to  the  negative  position,  they  were  found 

79 


MEMOIRS    ON 

to  be  about  8°  colder  than  the  air,  and  their  effect  on  the  arm 
seemed  now  to  diminish  on  standing,  instead  of  increasing,  as 
it  did  before  ;  as  the  arm  was  drawn  aside  about  2^  divisions 
less,  at  the  end  of  an  hour  after  the  motion  of  the  weights, 
than  it  was  at  first. 

It  seems  sufficiently  proved,  therefore,  that  the  effect  in  ques- 
tion is  produced,  as  above  explained,  by  the  difference  of  tem- 
perature between  the  weights  and  case ;  for  in  the  6th,  8th,  and 
9th*  experiments,  in  which  the  weights  were  not  much  warmer 
than  the  case,  their  effect  increased  but  little  on  standing  ; 
whereas,  it  increased  much,  when  they  were  much  warmer  than 
the  case,  and  -decreased  much,  when  they  were  much  cooler. 

It  must  be  observed,  that  in  this  apparatus,  the  box  in  which 
the  balls  play  is  pretty  deep,  and  the  balls  hang  near  the  bot- 
tom of  it,  which  makes  the  effect  of  the  current  of  air  more 
sensible  than  it  would  otherwise  be,  and  is  a  defect  which  I  in- 
tend to  rectify  in  some  future  experiments. 

EXPERIMENT  IX.     APRIL  29 

Weights  in  positive  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

34.7 

[ 

35. 

— 

— 

34.84 

34.65 

V 

"Weights  moved  to  negative  position 


23.8 

28 

29 

33.2 



29 

28 

23.9 

— 

32. 



24.15 

— 

31. 

— 

24.4 



30.4 

— 

28 

27 

24.7 

18'  29"  \ 

58  f 

— 

llh  18'  43' 

— 

28.52 

25  27  \ 

57  J 

— 

25  40 

28.25 

— 

28.01 



27.82 



27.63 

— 

27.55 

> 

— 

27.47 

7   4  ) 

53  f 

0   7  26 

Motion  of  arm        =6.32 
Time  of  vibration  =  6'  58" 


*  [This  is  evidently  a  misprint  for  6th,  7th,  and  8th.] 
80 


THE    LAWS    OF    GRAVITATION 


EXPERIMENT  X.     MAY  5 

Weights  in  positive  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

Difference 

34.5 

33.5 
34.4 

— 

— 

33.97 

Weights  moved  to  negative  position 


22.3 

28 
29 

10h  43'   42"  I 
44      6    J 

— 

10h  43'    36" 

33.2 

— 

— 

27.82 

—     . 

r    o" 

28 

27 

50    33    ) 
51      0    J 

— 

50    36 

22.6 

— 

— 

27.72 

32.5 

— 

— 

27.7 

23.2 

— 

— 

27.58 

31.45 

— 

— 

27.4. 

23.5 

— 

— 

27.28 

27 
28 

11    25    20    ) 

58    f 

— 

11    25    24 

30.7 

— 

— 

27.21 

— 

7      3 

28 
27 

32      0    ) 
32    40    J 

— 

32    27 

23.95 

— 

— 

27.21 

— 

6    56 

27 

39    19    ) 

on      .>•> 

28 

40      2    i" 

o»      &6 

30.25 

Motion  of  arm       =6.15 

Time  of  vibration  =  6'  59" 

EXPERIMENT  XL     MAY  6 

Weights  in  positive  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

34.9 

34.1 

— 

— 

34.47 

'      34.8 





34.49 

34.25 

Weights  moved  to  negative  position 

23.3 

28 
29 

9h   59'    59"  ) 

10      0    27    f 

— 

10h     0'     8" 

33.3 

— 

— 

28.42 

29 

6    52    ) 

27 

7    51    f 

— 

7      5 

23.8 

— 

— 

28.35 

F                                                                        81 

MEMOIRS    ON 


32.5 

— 

— 

28.3 

24.4 

missed 

24.8 

31.3 

— 



28.17 

29 

28 

10h  48'    37"  ) 
49    21    J 

— 

10h  49'     8" 

25.3 



— 

28.2 

28 
29 

56      8    I 
56    f 

— 

56    13 

30.9 

Motion  of  arm       =6.07 

Time  of  vibration  =  7'  1" 

In  the  three  foregoing  experiments,  the  index  was  purposely 
moved  so  that,  before  the  beginning  of  the  experiment,  the 
balls  rested  as  near  the  sides  of  the  case  as  they  could,  without 
danger  of  touching  it ;  for  it  must  be  observed,  that  when  the 
arm  is  at  35,  they  begin  to  touch.  In  the  two  following  ex- 
periments, the  index  was  in  its  usual  position. 


EXPERIMENT  XII.     MAY  9 

Weights*  in  negative  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

17.4 
17.4 
17.4 
17.4 

9h    45'       0" 

58      0 
10      8      0 
10      0 

17.4 

Weights  moved  to  positive  position 


28.85 

24 
22 

20    50    ) 
21    46    f 

— 

10h  20'   59" 

18.4 

— 

— 

23.49 

28.3 

— 

— 

23.57 

19.3 





23.67 

27.8 

— 

— 

23.72 

20. 

— 

— 

23.8 

27.4 

. 



23.83 

24 
23 

11      3    13    ) 
54    \ 

— 

11      3    14 

20.55 

— 

23.87 

23 
24 

9    45    ) 
10    28    f 

— 

10    18 

27. 

Motion  of  arm        =6.09 

Time  of  vibration  =  7'  3" 

82 


THE    LAWS    OF    GRAVITATION 
EXPERIMENT  XIII.     MAY  25 

Weights  in  negative  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

16. 

18.3 



17.2 

16.2 

Weights  moved 

to  positive  position 

29.6 

25 
24 

10h   22 

22"  I 
45    [ 

— 

10h  22'   56" 

17.4 

— 

— 

23.32 

23 
24 

29 
30 

59    ) 
23    f 

— 

30      3 

28.9 

— 

— 

23.4 

24 
23 

36 

37 

58   \ 
24   / 

— 

37      7 

18.4 

— 

— 

23.52 

23 
24 

44 

3    j. 
31    f 

— 

44    14 

28.4 

— 

— 

23.62 

19.3 





23.7 

27.8 

— 



23.7 

24 
23 

11       5 
6 

26    ) 

1    f 

— 

11      5    31 

19.9 

— 

— 

23.72 

23 
24 

12 

12    ) 
50    \ 

— 

12    35 

27.3 

Weights  moved 

to  negative  position 

13.5 

21.8 

— 

— 

17.75 

18 
17 

37 
38 

&\ 

— 

37    39 

13.9 

— 

— 

17.67 

17 

18 

44 
45 

*;} 

— 

44    45 

21.1 

— 

— 

17.62 

14.4 

— 



17.6 

20.5 

— 

— 

17.52 

14.7 

— 



17.47 

20. 

— 

— 

17.42 

18 
17 

0    19 
20 

57    I 
52    ] 

— 

0    20    24 

15. 

— 

— 

17.37 

17 
18 

27 
28 

15    I 
15    \ 

— 

27    30 

19.5 

Motion  of  the  arm  on  moving 

weights  from  —  to  +  =  6.12 

+  to  -=5.97 

Time  of  vibration  at  + 

=  7'  6" 

— 

=  7'  7" 

83 


MEMOIRS    ON 


EXPERIMENT  XIV.     MAY  26 

Weights  in  negative  position 


Extreme 

points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibnition 

16.1 
16.1 
16.1 
16.1 

9h    18'       0" 

24      0 
46      0 
49      0 

16.1 

Weights  moved  to  positive  position 


27.7 

23 
22 

10      0 
1 

46    \ 
16    f 

— 

10"   r    i" 

17.3 

— 

— 

22.37 

22 
23 

7 
8 

58    ) 

27    \ 

— 

8      5 

27.2 





22.5 

23 
22 

15 

2    i 
32    I 

— 

15      9 

18.3 



— 

22.65 

26.8 

'    , 



22.75 

19.1 

— 

— 

22.85 

26.4 

— 

— 

22.97 

23 

22 

43 
44 

40    ) 
22    \ 

— 

43    32 

20. 

— 

— 

23.15 

22 
23 

49 
50 

53    ) 
37    \ 

— 

50    41 

26.2 

Weights  moved  to  negative  position 

12.4 

16 
17 

11      7 

8 

53    ) 

27    \ 

— 

11      8    25 

21.5 





17.02 

17 
16 

15 
16 

30    I 
3    \ 

;',     — 

15    27 

12.7 





16.9 

20.7 

— 

— 

16.85 

13.3 

— 

— 

16.82 

20. 





16.72 

13.6 

— 



16.67 

16 
17 

50 
51 

33    ) 
19    \ 

— 

50    58 

19.5 

— 

— 

16.65 

17 
16 

57 
58 

53    ) 
44    } 

— 

58      6 

14. 

Motion  of  arm  by  moving 

weights  from  —  to  +  =  6.27 

+  to  -=6.13 

Time  of  vibration  at  + 

=  7'  6" 

- 

=  7'  6" 

84 


THE    LAWS    OF    GRAVITATION 

In  the  next  experiment,  the  balls,  before  the  motion  of  the 
weights,  were  made  to  rest  as  near  as  possible  to  the  sides  of 
the  case,  but  on  the  contrary  side  from  what  they  did  in  the 
9th,  10th,  and  llth  experiments. 

EXPERIMENT  XV.     MAY  27 

Weights  in  negative  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  ot 
vibration 

3.9 

3.85 



— 

3.61 

3.85 

— 

— 

3.61 

3.4 

Weights  moved  to  positive  position 

15.4 

10 
9 

10h    5'  59"  ) 
6  27    f 

— 

10h    5'  56" 

4.8 

'•    .      — 

— 

9.95 

9 

10 

12   43    ) 
13    11    f 

— 

13     5 

14.8 

— 

— 

10.07 

10 
9 

20   24    ) 

56    f 

— 

20  13 

5.9 

— 

— 

10.23 

14.35 

— 



10.35 

6.8 

— 

'  

10.46 

13.9 

— 

— 

10.52 

11 
10 

48   30    \ 
49    11    ] 

— 

48  42 

7.5 

— 

— 

10.6 

10 

11 

55   26    ) 
56   10    \ 

— 

55  48 

13.5 

Motion  of  the  arm  =  6.34 

Time  of  vibration  =  7'  7" 

The  two  following  experiments  were  made  by  Mr.  Gilpin, 

who  was  so  good  as  to  assist  me  on  the  occasion. 

EXPERIMENT  XVI.     MAY  28 

Weights  in  negative  position 

Extreme 
points 

Divisions 

Time 

Point  of 

rest 

Time  of  middle  of 
vibration 

22.55 

8.4 





15.09 

21. 

'  



14.9 

9.2 

85 


MEMOIRS    ON 

Weights  moved  to  positive  position 


26.6 

22 

21 

10h  22'  53"  ) 
23   20    I 

— 

10h  23'  15" 

15.8 





21. 

20 
21 

30     7    | 
36    f 

— 

30  30 

25.8 

— 

— 

21.05 

22 
21 

37   23    I 
55    \ 

— 

37  45 

16.8 



21.11 

20 
21 

44   29    ) 
45     4    f 

— 

45     1 

25.05 

— 

— 

21.11 

22 
21 

51    54    ) 
52   32    f 

— 

52  20 

17.57 

— 

— 

21.2 

21 

22 

59   31    | 
11     0    13    f 

— 

59  34 

24.6 

— 

— 

21.28 

22 

21 

6   24    } 

7     9    f 

— 

11     6  49 

18.3 

Motion  of  the  arm  =  6.1 

Time  of  vibration  =.  7'  16" 

EXPEKIMENT  XVII.     MAY  30 

Weights  in  negative  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

17.2 

10h   19'     0" 

17.1 

25     0 

17.07 

29     0 

17.15 

40     0 

17.45 

49     0 

17.42 

51     0 

17.42 

11      1     0 

17.42 

Weights  moved  to  positive  position 


28.8 

24 
23 

11  11  23  ) 
49  J 

— 

llh  11'  37" 

18.1 

— 

— 

23.2 

22 
23 

18  13  I 
43  \ 

— 

18  42 

27.8 

— 

23.12 

24 
23 

25  19  } 
49  f 

— 

25  40 

18.8 

— 

23.2 

23 
24 

32  41  ) 
33  13  \ 

— 

32  43 

86 

THE    LAWS    OF    GRAVITATION 


27.38 





23.31 

24 
23 

llh  39'  28"  f 
40     3    f 

— 

llh  39'  44" 

19.7 





23.44 

23 
24 

46   33    ) 

47    11    J 

— 

46  46 

27. 





23.52 

24 
23 

53   36    ) 
54  17    f 

— 

53  48 

20.4 





23.57 

23 
24 

0     0   34    ) 
1    18    ( 

— 

0     6  55 

26.5 

— 

— 

23.55 

24 
23 

7   34    ) 
8  21    \ 

— 

7  50 

20.8 

— 

— 

23.59 

23 
24 

14   30    ) 
15   24    f 

— 

14  58 

26.25 

Weights  moved  to  negative  position 

13.3 

17 

18 

32   19    ) 

48    f 

— 

32  44 

22.4 

— 

17.95 

18 
17 

39   46    > 
40   19    J 

— 

39  44 

13.7 

— 

— 

17.85 

17 

18 

46    26    ) 

47     0    f 

— 

46  48 

21.6 

— 



17.72 

18 
17 

53   43    ) 
54   20    f 

— 

53  50 

14. 

— 

— 

17.6 

17 

18 

1      0   39    I 

i  20  y 

— 

1     0  55 

20.8 

— 

— 

17.47 

18 
17 

7   39    | 
8   21    \ 

— 

7  59 

14.3 

— 

— 

17.37 

17 

14    54    ) 

18 

15   42    f 

— 

15     4 

20.1 

— 

— 

17.27 

18 
17 

21    32  ) 
22   22  J 

— 

22     5 

14.6 

Motion  of  the  arm  on  moving  weights  from  —  to  +  =  5.78 

+  to-  =5.64 

Time  of  vibration  at  4-                                                    =T  2" 

-                                                  =7'  3" 

87 


MEMOIRS    ON 

OK  THE  METHOD  OF  COMPUTING  THE  DENSITY  OF  THE  EARTH 
FROM  THESE  EXPERIMENTS 

I  shall  first  compute  this,  on  the  supposition  that  the  arm 
and  copper  rods  have  no  weight,  and  that  the  weights  exert  no 
sensible  attraction,  except  on  the  nearest  ball  ;  and  shall  then 
examine  what  corrections  are  necessary,  on  account  of  the  arm 
and  rods,  and  some  other  small  causes. 

The  first  thing  is,  to  find  the  force  required  to  draw  the  arm 
aside,  which,  as  was  before  said,  is  to  be  determined  by  the  time 
of  a  vibration. 

The  distance  of  the  centres  of  the  two  balls  from  each  other 
is  73.3  inches,  and  therefore  the  distance  of  each  from  the 
centre  of  motion  is  36.65,  and  the  length  of  a  pendulum  vi- 
brating seconds,  in  this  climate,  is  39.14;  therefore,  if  the 
stiffness  of  the  wire  by  which  the  arm  is  suspended  is  such, 
that  the  force  which  must  be  applied  to  each  ball,  in  order  to 
draw  the  arm  aside  by  the  angle  A,  is  to  the  weight  of  that 
ball  as  the  arch  of  A  to  the  radius,  the  arm  will  vibrate  in  the 
same  time  as  a  pendulum  whose  length  is  36.65  inches,  that  is,  in 

y  '  —    -  seconds  ;  and  therefore,  if  the  stiffness  of  the  wire  is 

such  as  to  make  it  vibrate  in  N  seconds,  the  force  which  must 
be  applied  to  each  ball,  in  order  to  draw  it  aside  by  the  angle 

A,  is  to  the  weight  of  the  ball  as  the  arch  of  Ax^pX.y^ 

-LAi  O  */•  XT: 

to  the  radius.  But  the  ivory  scale  at  the  end  of  the  arm  is 
38.3  inches  from  the  centre  of  motion,  and  each  division  is  -fa 
of  an  inch,  and  therefore  subtends  an  angle  at  the  centre,  whose 
arch  is  T|-g  ;  and  therefore  the  force  which  must  be  applied  to 
each  ball,  to  draw  the  arm  aside  by  one  division,  is  to  the  weight 

1         36.65  .  1 

of  the  ball  as  •  to  1,  or  as  to  1.* 


*  [Or  thus:  using  the  ordinary  notation  for  the  simple  pendulum  vibrating 
through  small  arcs,  if  the  force  on  each  ball  drawing  the  arm  aside  through  an 

arc  subtending  an  angle  of  A°  were  mg  x  ^     /   ,  the  arm  would  vibrate  like  a 

/Of*    £»  pr 

pendulum  of  the  same  length,  and  have  a  period  of\J  —   ~  seconds,  because  the 

*  oy.i4 

period  of  a  pendulum  varies  as  the  square  root  of  its  length.    But  the  force  varies 

as  -  ,  ;  therefore  the  force  required  to  draw  the  arm  through  A°  with 
(period}1  ' 


THE    LAWS    OF    GRAVITATION 


The  next  thing  is,  to  find  the  proportion  which  the  attraction 
of  the  weight  on  the  ball  hears  to  that  of  the  earth  thereon,  sup- 
posing the  ball  to  be  placed  in  the  middle  of  the  case,  that  is, 
to  be  not  nearer  to  one  side  than  the  other.  When  the  weights 
are  approached  to  the  balls,  their  centres  are  8.85  inches  from 
the  middle  line  of  the  case  ;  but,  through  inadvertence,  the 
distance,  from  each  other,  of  the  rods  which  support  these 
weights,  was  made  equal  to  the  distance  of  the  centres  of  the 
balls  from  each  other,  whereas  it  ought  to  have  been  somewhat 
greater.  In  consequence  of  this,  the  centres  of  the  weights  are 
not  exactly  opposite  to  those  of  the  balls,  when  they  are  ap- 
proached together;  and  the  effect  of  the  weights,  in  drawing 
the  arm  aside,  is  less  than  it  would  otherwise  have  been,  in  the 

o    OK 

triplicate  ratio  of '—  to  the  chord  of  the  an  He  whose  sine  is 

8  85  36'65 

-— ,  or  in  the  triplicate  ratio  of  the  cosine  of  i  this  angle  to 

OD.OD 

the  radius,  or  in  the  ratio  of  .9779  to  1.* 


period  N"  =  mg  x 


arc  of  Ac 


36.65 
X  8914 


oa  KK      "  on  1  A •  —  ^  !-     ^1^^  the  force  required  to  draw 

OO.DO  o".  14 

the  arm  through  1  scale  division  with  period  N" 
36.65     1 
38.3   '  20     36.65 
=  ^X-^65-X39Tl4^N 

1  36.65  1 

=  mg  x 


766N* 

*  [Let  W  be  the  position  of 
the  "weight  "  of  mass  W,  B 
the  position  it  was  intend- 
ed that  it  should  have,  and  m 
that  of  the  ' '  ball "  of  mass  m. 
The  distance  mE,  or  WA,  is 
8.85  inches,  and  OW  and  Om 
36.65  inches.  Call  WA  and 
Wm  a  and  b  respectively,  and 
G  the  gravitation  constant. 
Then  it  was  intended  that 
the  attraction  to  move  the  arm 

should  be    -      -2'm,  but   it  is 

G-W-m    a 

— 75 r  ;  a  nd  so  is  less  than 

b2         b 

was  intended  in  the  ratio  of 
—•to  1,  or  of  Cos*  ^  to  1.] 


39.14"  mgX  818 N2] 


89 


Fig.  d 


MEMOIRS    ON 

Each  o.f  the  weights  weighs  2,439,000  grams,  and  therefore  is 
equal  in  weight  to  10.64  spherical  feet  of  water  ;*  and  therefore 
its  attraction  on  a  particle  placed  at  the  centre  of  the  ball,  is  to 
the  attraction  of  a  spherical  foot  of  water  on  an  equal  particle 

(6    V 
s-svj    to  1.     The 
o.oO/ 

mean  diameter  of  the  earth  is  41,800,000  feet  ;f  and  therefore, 
if  the  mean  density  of  the  earth  is  to  that  of  water  as  D  to  one, 
the  attraction  of  the  leaden  weight  on  the  ball  will  be  to  that  of 

the  earth  thereon,  as  10.64  x  .9779  x  (—}    to  41,800,000  D  :  : 

\O.OO/ 

1  :  8, 739,000  D.t 

*  [That  is,  iff  equal  to  the  weight  of  a  sphere  of  water  which  can  be  inscribed  in 
a  cube  whose  volume  is  10.64  cu.  ft.,  or  we  can  express  the  volume  of  the  sphere 
by  the  number  10.64,  when  the  unit  of  volume  is  that  of  a  sphere  of  I  foot  in 

diameter,  that  is,  of—cu.ft  The  radius  of  a  spherical  foot  of  water  is,  accord- 
ingly, 6  inches.  Cavendish  evidently  uses  Kirwaris  estimate  0/253.35  grains 
to  the  cu.  in.  of  water. 

The  ensuing  calculation  can  be  stated  thus :  Call  d  and  d'  the  densities  of 
water  and  of  the  earth  respectively,  m  the  mass  of  the  ball,  and  G  the  gravita- 
tion constant.  The  volume  of  the  earth  in  spherical  units  is  (41  800  OOO)3,  and 
itsraditisQxll  800  000  inches. 

Gx  10.64  x 


Attraction  of  weight  on  ball  at  8.85  inches  _  (8.85)- 


x.9779 


Attraction  of  earth  on  ball  Gx  (41_800  000 ) J  x  d'  xwi 

"~(6x41  800l)()0)r 


.9779xl0.64x 


f— Y 

\8.8,V 


41  800  000  ^ 

1 


(I) 
(2) 


8  739  000  D    ' 

But  we  have  already  found  (page  89) 
Force  required  to  draw  the  arm  through  1  div.  _     1 
Weight  of  ball  '~818N»     ' 

Dividing  equation  (1)  by  (2)  we  have 

Attraction  of  weight  on  ball _____  _      818  N2  N2 

Force  required  to  draw  the  arm  through  1  div.  ~  8  739  GOOD  ~  10  683 D 
=  no.  of  div.  through  which  the  arm  io  drawn  EEB  div.] 

f  In  strictness,  we  ought,  instead  of  the  mean  diameter  of  the  earth,  to 
take  the  diameter  of  that  sphere  whose  attraction  is  equal  to  the  force  of 
gravity  in  this  climate;  but  the  difference  is  not  worth  regarding. 

\  [Ilutton  has  pointed  out  (54)  that  ihis  number  should  be  8,740,000 ;  but  it 
will  not  make  any  appreciable  change  in  the  value  of  D.] 

90 


THE    LAWS    OF    GRAVITATION 

It  is  shewn,  therefore,  that  the  force  which  must  be  applied 
to  each  ball,  in  order  to  draw  the  arm  one  division  out  of  its 

natural  position,  is         xra  of  the  weight  of  the  ball  ;  and,  if  the 

olo  JN 

mean  density  of  the  earth  is  to  that  of  water  as  D  to  1,  the  at- 

traction of  the  weight  on  the  ball  is      •  ^  of  the  weight  of 

o,  7  o",UUU  L) 

that  ball  ;  and  therefore  the  attraction  will  be  able  to  draw  the 
...  .  .      .         818  N2  N2 

arm  out  of  its  natural  position  by  or  *  dlV18' 


ions  ;  and  therefore,  if  on  moving  the  weights  from  the  mid- 
way to  a  near  position  the  arm  is  found  to  move  B  divisions,  or 
if  it  moves  2  B  divisions  on  moving  the  weights  from  one  near 
position  to  the  other,  it  follows  that  the  density  of  the  earth, 

_    .        N2 
°r  °   1S 


We  must  now  consider  the  corrections!  which  must  be  ap- 
plied to  this  result  ;  first,  for  the  effect  which  the  resistance  of 
the  arm  to  motion  has  on  the  time  of  the  vibration  :  2d,  for  the 
attraction  of  the  weights  on  the  arm  :  3d,  for  their  attraction 
on  the  farther  ball  :  4th,  for  the  attraction  of  the  copper  rods 
on  the  balls  and  arm  :  5th,  for  the  attraction  of  the  case  on  the 
balls  and  arm  :  and  6th,  for  the  alteration  of  the  attraction  of 
the  weights  on  the  balls,  according  to  the  position  of  the  arm, 
and  the  effect  which  that  has  on  the  time  of  vibration.  None 
of  these  corrections,  indeed,  except  the  last,  are  of  much  s4g- 
nification,  but  they  ought  not  entirely  to  be  neglected. 

As  to  the  first,  it  must  be  considered,  that  during  the  vibra- 
tions of  the  arm  and  balls,  part  of  the  force  is  spent  in  acceler- 
ating the  arm  ;  and  therefore,  in  order  to  find  the  force  re- 
quired to  draw  them  out  of  their  natural  position,  we  must 
find  the  proportion  which  the  forces  spent  in  accelerating  the 
arm  and  balls  bear  to  each  other. 

Let  EDCMc  (Fig.  4)  be  the  arm.  B  and  b  the  balls.  Gs 
the  suspending  wire.  The  arm  consists  of  4  parts  ;  first,  a  deal 
rod  Dcd,  73.3  inches  long  ;  2d,  the  silver  wire  DCW,  weighing  170 
grains  ;  3d,  the  end  pieces  DE  and  ed,  to  which  the  ivory 

*  [This  number  should  be  10,685.     See  last  note.} 

\  [For  a  discussion  of  these  corrections,  similar  to  that  of  Cavendish,  but 
with  modern  mathematical  treatment,  see  Reich  (67).] 

91 


MEMOIRS    ON 


Fig.   4 


vernier  is  fastened,  each  of  which  weighs  45  grains  ;  and  4th, 
some  brass  work  Cc,  at  the  centre.  The  deal  rod,  when  dry, 
weighs  2320  grains,  but  when  very  damp,  as  it  commonly  was 
during  the  experiments,  weighs  2400  ;  the  transverse  section  is 
of  the  shape  represented  in  Fig.  5  ;  the  thick- 
ness BA,  and  the  dimensions  of  the  part 
Dl&ed,  being  the  same  in  all  parts  ;  but  the 
breadth  B#  diminishes  gradually,  from  the 
middle  to  the  ends.  The  area  of  this  sec- 
tion is  .33  of  a  square  inch  at  the  middle,  and  .146  at  the 
end;  and  therefore,  if  any  point  x  (Fig.  4)  is  taken  in  cd, 

1  ex  .  .  ,       2400 x. 33 

and  —j  is   called  x,  this   rod  weighs  -^  Q  per   inch   at 


, 
cd 


73.3  x. 238 


,,         .,,.      2400X.146  2400     .33-. 184^ 

the  middle ;  -„-—  -  at  the  end,  and   -r— r  x  - 


73.3  x. 238 


73.3 


.238 


3320 -1848 


at  x;  and  therefore,  as  the  weight  of  the  wire  is 


7o.o 
per  inch,  the  deal  rod  and  wire  together  may  be  con- 


170 
7373 

.,    -  .  ,  ,  3490-1848  :r         .     . 

side  red  as  a  rod  whose  weight  at  x=  — • — ^-^ —  -  per  inch. 

I  O.O 

But  the  force  required  to  accelerate  any  quantity  of  matter 
placed  at  x,  is  proportional  to  x3 ;  that  is,  it  is  to  the  force  re- 
quired to  accelerate  the  same  quantity  of  matter  placed  at  d  as 
x*  to  1 ;  and  therefore,  if  cd  is  called  /,  and  x  is  supposed  to 
flow,  the  fluxion  of  the  force  required  to  accelerate  the  deal 

92 


THE    LAWS    OF    GRAVITATION 

x*lxx  (3490 -1848*)    . 
rod  and  wire  is  proportional  to  -          — -— - —         -i,  the  fluent 

Y  o.  o 

of  which,  generated  while  x  flows  from  c  to  d, 

=  1^3X 

so  that  the  force  required  to  accelerate  each. half  of  the  deal 
rod  and  wire,  is  the  same  as  is  required  to  accelerate  350  grains 
placed  at  d. 

The  resistance  to  motion  of  each  of  the  pieces  de,  is  equal  to 
that  of  48  grains  placed  at  d;  as  the  distance  of  their  centres  of 
gravity  from  C  is  38  inches.  The  resistance  of  the  brass  work 
at  the  centre  may  be  disregarded  ;  and  therefore  the  whole  force 
required  to  accelerate  the  arm,  is  the  same  as  that  required  to 
accelerate  398  grains  placed  at  each  of  the  points  D  and  d. 

Each  of  the  balls  weighs  11,262  grains,  and  they  are  placed 
at  the  same  distance  from  the  centre  as  D  and  d;  and  there- 
fore, the  force  required  to  accelerate  the  balls  and  arm  to- 
gether, is  the  same  as  if  each  ball  weighed  11,660,  and  the  arm 
had  no  weight ;  and  therefore,  supposing  the  time  of  a  vibra- 
tion to  be  given,  the  force  required  to  draw  the  arm  aside,  is 
greater  than  if  the  arm  had  no  weight,  in  the  proportion  of 
11,660  to  11,262,  or  of  1.0353  to  1. 

To  find  the  attraction  of  the  weights  on  the  arm,  through  d 
draw  the  vertical  plane  dtvb  perpendicular  to  Dd,  and  let  w 
be  the  centre  of  the  weight,  which,  though  not  accurately  in 
this  plane,  may,  without  sensible  error,  be  considered  as  placed 
therein,  and  let  b  be  the  centre  of  the  ball ;  then  wb  is  hori- 
zontal andr=8.85,  and  db  is  vertical  and=5.5;  let  wd=a,  wb 

dx 
=  b,  and  let  -=-,  or  I  —  x,  =z;  then  the  attraction  of  the  weight 

on  a  particle  of  matter  at  *,  in  the  direction  bw,  is  to  its  at- 
traction on  the  same  particle  placed  at  b:: b3 :  (#2-f-zT)*,  or  is 

b3 
proportional  to  -          — 3,  and  the  force  of  that  attraction  to 

53  x  (1  — z) 
move  the  arm,  is  proportional  to  a  a  ^,  and  the  weight  of 

the  deal  rod  and  wire  at  the  point  x,  was  before  said  to  be 

3490  _ 1 848  z     1 642  + 1 848  z 

— — = ;— —       per    inch;    and    therefore,  if  dx 

<o.o  to.o 

flows,  the  fluxion  of  the  power  to  move  the  arm 

93 


73.3 


MEMOIRS    ON 
~r%^=*x  (821+924 


\          i 

w*^ 


;  which,  as  -  =  .08, 


924  *'«          ,r,     fl 
.  The  fluent  of  this 


895  £32  103  63         103  £3     924 


and  the  force  with  which  the  attraction  of  the  weight,  on  the 
nearest  half  of  the  deal  rod  and  wire,  tends  to  move  the  arm, 
is  proportional  to  this  fluent  generated  while  z  flows  from  0 
to  1,  that  is,  to  128  grains. 

The  force  with  which  the  attraction  of  the  weight  on  the  end 

piece  de  tends  to  move  the  arm,  is  proportional  to  47  x  —3  [approxi- 
mately], or  29  grains;  and  therefore  the  whole  power  of  the 
weight  to  move  the  arm,  by  means  of  its  attraction  on  the  near- 
est part  thereof,  is  equal  to  its  attraction  on  157  grains  placed 

157 
at#,  which  is  ,  or  .0139  of  its  attraction  on  the  ball.* 

11/CD/O 

It  must  be  observed,  that  the  effect  of  .the  attraction  of  the 
weight  on  the  whole  arm  is  rather  less  than  this,  as  its  attrac- 
tion on  the  farther  half  draws  it  the  contrary  way ;  but,  as  the 
attraction  on  this  is  small,  in  comparison  of  its  attraction  on 
the  nearer  half,  it  may  be  disregarded. 

The  attraction  of  the  weight  on  the  furthest  ball,  in  the 
direction  btv,  is  to  its  attraction  on  the  nearest  ball  ::  wb3 :  wB3f- 
::. 0017:1;  and  therefore  the  effect  of  the  attraction  of  the 
weight  on  both  balls,  is  to  that  of  its  attraction  on  the  nearest 
ball::. 9983  : 1. 


*  [A  few  minor  misprints  in  Hie  last  two  paragraphs  in  the  original  paper 
have  been  corrected.  A  recalculation  seems  to  give  142.5  instead  ofl2S,  and 
28  instead  of  29,  grains;  this  would  change  the  value  ofD  by  1  part  in  1000.] 

f  [This  is  erroneously  printed  in  the  original  as  wd3 :  wDs.~\ 

94 


THE    LAWS    OF    GRAVITATION 


To  find  the  attraction  of  the  copper  rod  on  the  nearest  ball, 
let  1)  and  w  (Fig.  6)  be  the  centres  of  the  ball  and  weight,  and 
ea  the  perpendicular  part  of  the  copper  rod,  which  consists  of 
two  parts,  ad  and  de.  ad  weighs  22,000  grains,  and  is  16  inches 
long,  and  is  nearly  bisected  by  w.  de  weighs  41,000,  and  is  46 
inches  long,  ivb  is  8.85  inches,  and  is  perpendicular  to  ew. 
Now,  the  attraction  of  a  line  ew, 
of  uniform  thickness,  on  b,  in  the 
direction  bw,  is  to  that  of  the  same 
quantity  of  matter  placed  at  w\\  bw 
:eb;  and  therefore  the  attraction 
of  the  part  da  equals  that  of 

22,000  x  wb 

,  or  16,300,  placed  at  w; 


db 
and    the 


attraction    of    de   equals 


Fig.  6 


that   of   41,000x^x^-41,000 
ed     be 

dw     bw  ,, 

X— yX-r-i i,  or  2500,  placed  at  the 

ed     bd 

same  point ;  so  that  the  attraction 

of  the  perpendicular  part  of  the   w 

copper  rod  on  b,  is  to  that  of  the 

weight  thereon,  as  18,800  :  2,439,- 

000,  or  as  .00771  to  1.     As  for  the 

attraction  of  the  inclined  part  of 

the  rod  and  wooden  bar,  marked 

Pr  and  rr  in  Fig.  1,  it  may  safely 

be  neglected,  and  so  may  the  attraction  of  the  whole  rod  on  the 

arm  and  farthest  ball ;  and  therefore  the  attraction  of  the  weight 

and  copper  rod,  on  the  arm  and  both  balls  together,  exceeds  the 

attraction  of  the  weight  on  the  nearest  ball,  in  the  proportion 

of  .9983  + .0139  +  . 0077  to  one,  or  of  1.0199  to  1. 

The  next  thing  to  be  considered,  is  the  attraction  of  the  ma- 
hogany case.  Now  it  is  evident,  that  when  the  arm  stands  at 
the  middle  division,  the  attractions  of  the  opposite  sides  of  the 
case  balance  each  other,  and  have  no  power  to  draw  the  arm 
either  way.  When  the  arm  is  removed  from  this  division,  it  is 
attracted  a  little  towards  the  nearest  side,  so  that  the  force  re- 
quired to  draw  the  arm  aside  is  rather  less  than  it  would  other- 
wise be  ;  but  yet,  if  this  force  is  proportional  to  the  distance  of 
the  arm  from  the  middle  division,  it  makes  no  error  in  the  re- 

95 


MEMOIRS    ON 

suit  ;  for,  though  the  attraction  will  draw  the  arm  aside  more 
than  it  would  otherwise  do,  yet,  as  the  accelerating  force  by 
which  the  arm  is  made  to  vibrate  is  diminished  in  the  same 
proportion,  the  square  of  the  time  of  a  vibration  will  be  in- 
creased in  the  same  proportion  as  the  space  by  which  the  arm 
is  drawn  aside,  and  therefore  the  result  will  be  the  same  as  if 
the  case  exerted  no  attraction  ;  but,  if  the  attraction  of  the 
case  is  not  proportional  to  the  distance  of  the  arm  from  the 
middle  point,  the  ratio  in  which  the  accelerating  force  is  di- 
minished is  diiferent  in  different  parts  of  the  vibration,  and 
the  square  of  the  time  of  a  vibration  will  not  be  increased  in 
the  same  proportion  as  the  quantity  by  which  the  arm  is  drawn 
aside,  and  therefore  the  result  will  be  altered  thereby. 

On  computation,  I  find  that  the  force  by  which  the  attrac- 
tion draws  the  arm  from  the  centre  is  far  from  being  propor- 
tional to  the  distance,  but  the  whole  force  is  so  small  as  not  to 
be  worth  regarding;  for,  in  no  position  of  the  arm  does  the 
attraction  of  the  case  on  the  balls  exceed  that  of  ^th  of  a  spheric 
inch  of  water,  placed  at  the  distance  of  one  inch  from  the  cen- 
tre of  -the  balls  ;  and  the  attraction  of  the  leaden  weight  equals 
that  of  10.6  spheric  feet  of  water  placed  at  8.85  inches,  or  of 
234  spheric  inches  placed  at  1  inch  distance  ;  so  that  the  at- 
traction of  the  case  on  the  balls  can  in  no  position  of  the  arm 
exceed  TTYo  of  that  of  the  weight.  The  computation  is  given 
hi  the  Appendix. 

It  has  been  shown,  therefore,  that  the  force  required  to  draw 
the  arm  aside  one  division,  is  greater  than  it  would  be  if  the 
arm  had  no  weight,  in  the  ratio  of  1.0353  to  1,  and  therefore 

7T-rTa  of  the  weight  of  the  ball  ;  and  moreover,  the  attraction 

818  JN 

of  the  weight  and  copper  rod  on  the  arm  and  both  balls  to- 
gether, exceeds  the  attraction  of  the  weight  on  the  nearest  ball, 

1  0199 
in  the  ratio  of  1.0199  to  1,  and  therefore—  —  rr  of  the 


weight  of  the  ball  ;  consequently  D  is  really  equal  to  - 
I  0199  N2  N2  i* 

iMtead  °f 


mer  computation.     It  remains  to  be  considered  how  much  this 
is  affected  by  the  position  of  the  arm. 

*  [This  should  be  10,846  ;  see  note  on  pp.  90  and  91.] 
96 


THE    LAWS    OF    GRAVITATION 

Suppose  the  weights  to  be  approached  to  the  balls  ;  let  W 
(Fig.  7)  be  the  centre  of  one  of  the  weights;  let  M  be  the  cen- 
tre of  the  nearest  ball  at  its  mean  position,  as  when  the  arrn  is 
at  20  divisions  ;  let  B  be  the  point  which  it  actually  rests  at  ; 
and  let  A  be  the  point  which  it  would  rest  at,  if  the  weight  was 
removed  ;  consequently,  AB  is  the  space  by  which  it  is  drawn 
aside  by  means  of  the  attraction  ;  and  let  M/3  be  the  space  by 
which  it  would  be  drawn  aside,  if  the  attraction  on  it  was 


Fig.l 

the  same  as  when  it  is  at  M.     But  the  attraction  at  B  is  greater 
than  at  M,  in  the  proportion  of  WM2  :  WB2  ;  and   therefore, 

AT3     ,ra     WM2     ...     /t     2MB\ 

AB  =  M/3  x  ^gj  =  M/3  x  \l  +  ^^  j  ,  very  nearly. 


Let  now  the  weights  be  moved  to  the  contrary  near  position, 
and  let  w  be  now  the  centre  of  the  nearest  weight,  and  b  the  point 


of  rest  of  the  centre  of  the  ball  ;  then  M  =  Mft  x  M  -f 

and  B&^M3xH-+2M3xl+;  so  that  the 


whole  motion  B&  is  greater  than  it  would  be  if  the  attraction 
on  the  ball  was  the  same  in  all  places  as  it  is  at  M,  in  the  ratio 

of  14-  to  one  ;  and,  therefore,  does  not  depend   sensibly 

on  the  place  of  the  arm,  in  either  position  of  the  weights,  but 
only  on  the  quantity  of  its  motion,  by  moving  them. 

This  variation  in  the  attraction  of  the  weight,  affects  also 
the  time  of  vibration  ;  for,  suppose  the  weights  to  be  ap- 
proached to  the  balls,  let  W  be  the  centre  of  the  nearest 
weight  ;  let  B  and  A  represent  the  same  things  as  before  ;  and 
let  x  be  the  centre  of  the  ball,  at  any  point  of  its  vibration  ;  let 
AB  represent  the  force  with  which  the  ball,  when  placed  at  B, 
is  drawn  towards  A  by  the  stiffness  of  the  wire  ;  then,  as  B  is 
the  point  of  rest,  the  attraction  of  the  weight  thereon  will  also 
equal  AB  ;  and,  when  the  ball  is  at  x,  the  force  with  which  it 
is  drawn  towards  A,  by  the  stiffness  of  the  wire,  =  Ao?,  and  that 
with  which  it  is  drawn  in  the  contrary  direction,  by  the  attrac- 

WB2 

tion,=AB  x  ^-^  ;  so  that  the  actual  force  by  which  it  is  drawn 

W  x 
G  97 


MEMOIRS    ON 

ABxWB2  /,      2Bz\ 

towards    A=Az—  —  ^—^  —  =  AB  +  Rx—  ABx  M  +^^Jr=Bic 

2B#  x  AB 

nearty*     ^°  ^at  the  ac^ua^  force  with  which 


WR 

the  ball  is  drawn  towards  the  middle  point  of  the  vibration,  is 
less  than  it  would  be  if  the  weights  were  removed,  in  the  ratio 

2AB 
of  1  —  TiTTr  to  one,  and  the  square  of  the  time  of  a  vibration  is 

2AB 

increased  in  the  ratio  of  1  to  1  —  w^  ;  which  differs  very  little 


from  that  of   l  +  TT?r  to  1,  which  is  the  ratio  in  which  the 


motion  of  the  arm,  by  moving  the  weights  from  one  near  posi- 
tion to  the  other,  is  increased. 

The  motion  of  the  ball  answering  to  one  division  of  the  arm 

36  65* 
=o?  ~QO~Q  5  and>  ^  -M-^t  is  the  motion  of  the  ball  answering 

20  X  uo.O 

,  ..  .  .  MB  36.65<Z  d 

to  d  dmsions  on  the  arm,   _=_-.=_  ;   and 


therefore,  the  time  of  vibration,  and  motion  of  the  arm,  must 

rrected  as  follows  : 

If  the  time  of  vibration  is  determined  by  an  experiment  in 
which  the  weights  are  in  the  near  position,  and  the  motion  of 
the  arm,  by  moving  the  weights  from  the  near  to  the  midway 
position,  is  d  divisions,  the  observed  time  must  be  diminished 

in  the  subduplicate  ratio  of  1  —  -^-  to  1,  that  is,  in  the  ratio 

-LoO 

of  1  --  *—  to  1;  but,  when  it  is  determined  by  an  experiment 
185 

in  which  the  weights  are  in  the  midway  position,  no  correction 
must  be  applied. 

To  correct  the  motion  of  the  arm  caused  by  moving  the 
weights  from  a  near  to  the  midway  position,  or  the  reverse, 
observe  how  much  the  position  of  the  arm  differs  from  20 
divisions,  when  the  weights  are  in  the  near  position  ;  let  this 
be  n  divisions,  then,  if  the  arm  at  that  time  is  on  the  same  side 
of  the  division  of  20  as  the  weight,  the  observed  motion  must  be 

*  [This  number,  36.65,  Iwre,  and  again  in  the  next  line,  is  erroneously 
printed  in  Cavendish's  memoir  as  36.35.] 
f  [In  the  original  this  is  erroneously  printed  as  mB.] 


THE    LAWS    OF    GRAVITATION 


diminished  by  the  :—  part  of  the  whole ;  but,  otherwise,  it 

loO 

must  be  as  much  increased. 

If  the  weights  are  moved  from  one  near  position  to  the  other, 
and  the  motion  of  the  arm  is  2d  divisions,  the  observed  motion 

must  be  diminished  by  the  — — :  part  of  the  whole. 

185 

If  the  weights  are  moved  from  one  near  position  to  the  other, 
and  the  time  of  vibration  is  determined  while  the  weights  are 
in  one  of  those  positions,  there  is  no  need  of  correcting  either 
the  motion  of  the  arm,  or  the  time  of  vibration.^/ 


T 


CONCLUSION 


The  following  table  contains  the  result  of  the  experiments 


Exper. 

Mot.  weight 

Mot.  arm 

Do.  corr. 

Time  vib. 

Do.  corr. 

Density 

1  j 

m.  to  4- 

14.32 

13.42 



5.5 

H 

+  to  m. 

14.1 

13.17 

14'    55" 

— 

5.61 

oj 

m.  to  + 

15.87 

14.69 

— 

— 

4.88 

2i 

+  to  m. 

15.45 

14.14 

14    42 

— 

5.07 

q5 

+  to  m. 

15.22 

13.56 

14    39 

— 

5.26 

3i 

m.  to  + 

14.5 

13.28 

14    54 

— 

5.55 

I 

m.  to  + 

3.1 

2.95 

6'    54" 

5.36 

4\ 

+  to  - 

6.18 

— 

7      1 

— 

5.29 

} 

-  to  4 

5.92 

— 

7      3 

— 

5.58 

KJ 

+  to  - 

5.9 

— 

7      5 

— 

5.65 

5i 

-  to  + 

5.98 

— 

7      5 

— 

5.57 

«j 

m.  to  — 

3.03 

2.9  1 

5.53 

6i 

-  to  + 

5.9 

5.71 

7        A 

5.62 

7i 

m.  to  — 

-  to  + 

3.15 
6.1 

3.03  1 
5.9    f 

i        1 

by 

6    57 

5.29 
5.44 

m.  to  — 

3.13 

3.00  | 

mean 

5.34 

• 

-   to  + 

5.72 

5.54J 

5.79 

9 

+  to  - 

6.32 

— 

6    58 

— 

5.1 

10 

+  to  - 

6.15 

, 

6    59 



5.27 

11 

-f  -to  — 

6.07 

— 

7      1 

— 

5.39 

12 

-  to  + 

6.09 



7      3 



5.42 

-  to  + 

6.12 



7      6 

-  

5.47 

13 

+  to  - 

5.97 

— 

7      7 

— 

5.63 

—  to  + 

6.27 



7      6 



5.34 

14 

+  to  - 

6.13 

— 

7      6 

— 

5.46 

15 

-  to  + 

6.34 

— 

7      7 

— 

5.3 

16 

-  to  + 

6.1 

— 

7    16 

— 

5.75 

17J 

-  to  + 

5.78 

— 

7      2 

— 

5.68 

17  j 

+  to  - 

5.64 

— 

7      3 

— 

5.85 

*  [  The  corrections  neutralize  each  other,  since  they  are  the  same  for  N2  and 
B,  whose  ratio  enters  into  the  expression  for  D.J 


MEMOIRS    ON 

From  this  table  it  appears,  that  though  the  experiments 
agree  pretty  well  together,  yet  the  difference  between  them, 
both  in  the  quantity  of  motion  of  the  arm  and  in  the  time  of 
vibration,  is  greater  than  can  proceed  merely  from  the  error  of 
observation.  As  to  the  difference  in  the  motion  of  the  arm,  it 
may  very  well  be  accounted  for,  from  the  current  of  air  pro- 
duced by  the  difference  of  temperature  ;  but,  whether  this  can 
account  for  the  difference  in  the  time  of  vibration,  is  doubtful. 
If  the  current  of  air  was  regular,  and  of  the  same  swiftness  in 
all  parts  of  the  vibration  of  the  ball,  I  think  it  could  not ;  but, 
as  there  will  most  likely  be  much  irregularity  in  the  current,  it 
may  very  likely  be  sufficient  to  account  for  the  difference. 

By  a  mean  of  the  experiments  made  with  the  wire  first  used, 
the  density  of  the  earth  comes  out  5.48*  times  greater  than 
that  of  water ;  and  by  a  mean  of  those  made  with  the  latter 
wire,  it  comes  out  the  same  ;  and  the  extreme  difference  of  the 
results  of  the  23  observations  made  with  this  wire,  is  only  .75  ; 
so  that  the  extreme  results  do  not  differ  from  the  mean  by  more 
than  .38,  or  ^  of  the  whole,  and  therefore  the  density  should 
seem  to  be  determined  hereby,  to  great  exactness.  It,  indeed, 
may  be  objected,  .that  as  the  result  appears  to  be  influenced  by 
the  current  of  air,  or  some  other  cause,  the  laws  of  which  we 
are  not  well  acquainted  with,  this  cause  may  perhaps  act  al- 
ways, or  commonly,  in  the  same  direction,  and  thereby  make  a 
considerable  error  in  the  result.  But  yet,  as  the  experiments 
were  tried  in  various  weathers,  and  with  considerable  variety 
in  the  difference  of  temperature  of  the  weights  and  air,  and 
with  the  arm  resting  at  different  distances  from  the  sides  of 
the  case,  it  seems  very  unlikely  that  this  cause  should  act  so 
uniformly  in  the  same  way,  as  to  make  the  error  of  the  mean 
result  nearly  equal  to  the  difference  between  this  and  the  ex- 

*  [  This  should  be  5. 31.  Had  the  third  number  in  the  column  of  densities  been 
5.88,  instead  of  4. 88,  the  average  would  have  been  as  Cavendish  gave  it.  But 
Baity  (79,  p.  90)  recalculated  the  densities  from  Cavendish's  data,  and  found 
4.88  to  be  correct.  Curiously  enough  Cavendish  made  the  same  error  in  deduc- 
ing the  mean  result  of  the  whole  number  of  experiments.  It  should  be  5.448,  not 
5.48  (which  would  be  had  by  putting  5.88  in  place  0/4.88),  with  a  probable  error 
of  .033.  The  mean  result  of  the  last  23  observations  is  5.48.  The  greatest  dif- 
ference of  the  single  results  from  one  another  is  .  97  ;  and  the  extreme  result 
differs  from  the  mean  of  all  by  .57,  or  fa  of  the  whole.  For  an  account  of 
Buttons  reflections  on  Cavendish's  "pretty  and  amusing  little  experiment " 
see  his  paper  (45  and  54) ;  they  are  referred  to  in  this  volume  on  page  105.1 

100 


T  11 E    LA  W  S    OF  :GJi  A  V,  IT  £/f  I O  N 


treme  ;  and,  therefore,  it  seems  very  unlikely  that  the  density 
of  the  earth  should  differ  from  5.48  by  so  much  as  -fa  of  the 
whole.* 

Another  objection,  perhaps,  may  be  made  to  these  experi- 
ments, namely,  that  it  is  uncertain,  whether,  in  these  small 
distances,  the  force  of  gravity  follows  exactly  the  same  law  as 
in  greater  distances.  There  is  no  reason,  however,  to  think 
that  any  irregularity  of  this  kind  takes  place,  until  the  bodies 
come  within  the  action  of  what  is  called  the  attraction  of  co- 
hesion, and  which,  seems  to  extend  only  to  very  minute  dis- 
tances. With  a  view  to  see  whether  the  result  could  be  affected 
by  this  attraction,  I  made  the  9th,  10th,  llth,  and  15th  experi- 
ments, in  which  the  balls  were  made  to  rest  as  close  to  the  sides 
of  the  case  as  they  could  ;  but  there  is  no  difference  to  be  de- 
pended on,  between  the  results  under  that  circumstance,  and 
when  the  balls  are  placed  in  any  other  part  of  the  case. 

According  to  the  experiments  made  by  Dr.  Maskelyne,  on 
the  attraction  of  the  hill  Schehallien,  the  density  of  the  earth 
is  4£  times  that  of  water ;  which  differs  rather  more  from  the 
preceding  determination  than  I  should  have  expected.  But  I 
forbear  entering  into  any  consideration  of  which  determination 
is  most  to  be  depended  on,  till  I  have  examined  more  carefully 
how  much  the  preceding  determination  is  affected  by  irregular- 
ities whose  quantity  I  cannot  measure. 

*  [See  note  on  page  100.] 
101 


APPENDIX 


ON   THE   ATTRACTION   OF  THE   MAHOGANY  CASE  ON  THE   BALLS 

THE  first  thing  is,  to  find  the  attraction  of  the  rectangular 
fc  plane  ck'pb  (Fig  8)  on  the 

point  a,  placed  in  the  line 
ac  perpendicular  to  this 
plane. 

Let  ac  —  a,  ck  —  b,  cb  — 

x,  and  let 
V 


a2  -h  a*a 

traction  of  the  line  bfl  on 

a,    in   the   direction   ab,  = 

— — - — „  •   and  therefore,  if 
*>    ab  x  afi> 

cb  flows,  the  fluxion  of  the 
attraction  of  the  plane  on  the  point  a,  in  the  direction  cb,  = 
bx  x  —  bw  —  bw 


T  X 


=  v1 ',   then   the   at- 


Fig.  8 


V*1  +  ^ 

v          w 


:,  the  variable  part  of  the  fluent  of  which  =  —  log  (v 


v)2,  and  therefore  the  whole  attractio 


ck  4-  ak 


ion  =  log  ( 
-,     .  y    ac 

-^-  —  ~\  •  so  that  the  attraction  of  the  plane,  in  the  direction 
bfi+afi) 

cb,  is  found  readily  by  logarithms,  but  I  know  no  way  of  finding 
its  attraction  in  the  direction  ac,  except  by  an  infinite  series.* 


*  \Playfair  has  given  an  expression  in  finite,  terms  for  this  attraction  on  pp. 
225-8  of  his  paper  in  the  Trans.  Roy.  Soc.  Edin.,  vol.  6,  1812,  pp.  187-243, 

102 


MEMOIRS  ON  THE  LAWS  OF  GRAVITATION 
The  two  most  convenient  series  I  know,  are  the  following  : 
First  Series.  Let  —  =  TT,  and  let  A=arc  whose  tang,  is  TT,  B 

7T3  7T5 

=  A—  TT,  C  =  B+-5~  ,  D  =  C—  —  -,  etc.     Then  the  attraction  in  the 
o  o 


_  _      /  ,      B?tf2     3Cw4      3-5Dw6         \  * 
direction  ac=  y'l—w*  x  I  A  +  -%~  +  -%r±  +"2T4~-  6  '   .     /' 

For  the  second  series,  let  A=arc  whose  tang.  =  —  ,  B=:A  —  , 

7T  7T 

,  D  =  C—  -£-=,  etc.      Then  the  attraction  =  a  re.  90° 

7TJ  O7T5 


It  must  be  observed,  that  the  first  series  fails  when  IT  is 
greater  than  unity,  and  the  second,  when  it  is  less  ;  but,  if  b  is 
taken  equal  to  the  least  of  the  two  lines  ck  and  cb,  there  is  no 
case  in  which  one  or  the  other  of  them  may  not  be  used  con- 
veniently. 

By  the  help  of  these  series,  I  computed  the  following  table  : 


.1962 

.3714 

.5145 

.6248 

.7071 

.7808 

.8575 

.9285 

.9815 

i. 

.1962 

.00001 

.3714 

.00039 

.00148 

.5145 

.00074 

.00277 

.00521 

.6248 

00110 

.00406 

.00778 

.01183 

.7071 

00140 

.00522 

.01008 

.01525 

.02002 

.7808 

.00171 

.00637 

.01245 

.01  896!.  02405 

03247 

.8575 

00207 

00772 

.01522 

02339.03116 

.03964 

.05057 

.9285 

00244 

.00910 

.01810 

.02807.  03778!.  04867 

06319 

.08119 

.9815 

00271 

.01019 

.02084 

.03193 

.04868.05639 

07478 

09931 

.12849 

1. 

00284 

.01054 

.02135 

.03347 

.045601.05975 

.07978 

.10789 

.14632 

.19612 

ck 

Find  in  this  table,  with  the  argument  —7  at  top,  and  the  ar- 

ctfc 

cb 
gurnent  -    in  the  left-hand  column,  the  corresponding  logarithm ; 


entitled  "  Of  the  Solids  of  Greatest  Attraction,  or  those  which,  among  all  the 
8f>lirl&  that  have  certain  Properties,  attract  with  the  greatest  Force  in  a  given 
Direction."] 

*  [In  the  last  term  of  the  series  the  coefficient  D  was  omitted  in  the  original.] 

103 


MEMOIRS    ON 


then  add  together  this  logarithm,  the  logarithm  of  -j-t  and  the 

cb  afc 

logarithm  of  —;  the  sum  is  the  logarithm  of  the  attraction. 

To  compute  from  hence  the  attraction  of  the  case  on  the 
A  E  ball,  let  the  box  DCBA  (Fig. 

1),  in  which  the  ball  plaj^s,  be 
divided  into  two  parts,  by  a 
vertical  section,  perpendic- 
ular to  the  length  of  the 
case,  and  passing  through 
the  centre  of  the  ball ;  and, 
in  Fig.  9,  let  the  parallel- 
epiped AEDEaMe  be  one  of 
these  parts,  ABDE  being  the 
above-mentioned  vertical  sec- 
tion ;  let  x  be  the  centre  of 
the  ball,  and  draw  the  par- 
allelogram pnpmlx  parallel  to 
and  xgrp  parallel  to 
and  bisect  /3S  in  c. 
Now,  the  dimensions  of  the 
box,  on  the  inside,  are  Bft=1.75  ;  BD=3.6;  B/3=1.75;  and 
f)A  —  5 ;  whence  I  find  that,  if  xc  and  fix  are  taken  as  in  the  two 
upper  lines  of  the  following  table,  the  attractions  of  the  differ- 
ent parts  are  as  set  down  below. 


Fig.  9 


Excess  of  attraction 


Sum  of  these 
Excess  of  attraction 


xc 
(3x 

of  T>drg  above  Bbrg 
mdrp  above  nbrp 
mesp  above  nasp 


of  Bbnp  above  Ddmd 
Aanfl  above  JZemd 

Whole  attraction  of  the  inside  surface  of  the  ) 
half  box.  j" 


.75 
1.05 
.2374 
.2374 
.3705 

.5 

1.3 
.1614 
.1614 
.2516 

.25 

1.55 
.0813 
.0813 
.1271 

.8453 
.5007 
.4677 

.5744 
.3271 
.3079 

.2897 
.1606 
.1525 

.1231 

.0606 

.0234 

It  appears,  therefore,  that  the  attraction  of  the  box  on  x  in- 
creases faster  than  in  proportion  to  the  distance  xc. 

The  specific  gravity  of  the  wood  used  in  this  case  is  .61,  and 
its  thickness  is  f  of  an  inch  ;  and  therefore,  if  the  attraction 
of  the  outside  surface  of  the  box  was  the  same  as  that  of  the 

104 


THE    LAWS    OF    GRAVITATION 

inside,  the  whole  attraction  of  the  box  on  the  ball,  when  ex 
=  .75,  would  be  equal  to  2  x.1231  x.61  xf  cubic  inches,  or  .201 
spheric  inches  of  water,  placed  at  the  distance  of  one  inch  from 
the  centre  of  the  ball.  In  reality  it  can  never  be  so  great  as 
this,  as  the  attraction  of  the  outside  surface  is  rather  less  than 
that  of  the  inside  ;  and,  moreover,  the  distance  of  x  from  c  can 
never  be  quite  so  great  as  .75  of  an  inch,  as  the  greatest  motion 
of  the  arm  is  only  1|  inch. 


Much  has  been  written  concerning  the  Cavendish  experi- 
ment ;  the  following  references  may  be  consulted  to  advantage. 

Gilbert  (40),  in  1799,  translated  the  greater  part  of  Cavendish's 
paper  into  German  for  his  Annalen,  adding  many  explanatory 
notes.  A  few  years  later  Brand es  (42)  gave  a  fresh  mathemati- 
cal analysis  of  the  experiment,  including  the  equations  for  the 
time  of  swing  of  the  torsion  pendulum  in  the  experiment  pro- 
posed by  Muncke  (see  below).  In  1815,  the  original  paper  of 
Cavendish  was  translated  entire  into  French  by  M.  Chompre 
(50). 

In  1821,  Hutton  (54)  recalculated  the  results  of  the  experi- 
ment after  Cavendish's  own  formulae,  and  found,  as  he  thought, 
a  "copious  list  of  errata,  some  of  which  are  large  or  import- 
ant/' The  mean  of  the  first  6  experiments  so  corrected  is  5.19, 
and  of  the  other  23  is  5.43  ;  the  mean  of  these  two  means  is 
5.31,  which  Hutton  takes  as  the  correct  result  given  by  the 
Cavendish  experiment.  Baily  states,  however  (79,  pp.  92-96), 
that  Hutton  himself  had  fallen  into  error,  and  that  the  com- 
putations of. Cavendish  are  correct  except  in  the  one  detail  re- 
ferred to  on  page  100  of  this  volume.  Baily  gives  a  very  care- 
ful criticism  of  the  experiment  on  pp.  88-91  of  his  memoir.  He 
remarks  that  "  Cavendish's  object,  in  drawing  up  his  memoir, 
appears  to  have  been  more  for  the  purpose  of  exhibiting  a 
specimen  of  what  he  considered  to  be  an  excellent  method  of 
determining  this  important  inquiry,  than  of  deducing  a  result 
that  should  lay  claim  to  the  full  confidence  of  the  scientific 
world."  Baily  points  out  that  the  time  was  not  determined 
with  due  accuracy ;  that  the  experiments  were  not  arranged  in 
groups,  in  order  to  eliminate  the  error  arising  from  the  march 
of  the  resting  point;  and  that  the  distance  between  the  weight 

105 


MEMOIRS    ON 

and  the  ball  was  assumed  constant.  We  shall  see  later  from 
the  accounts  of  the  investigations  of  Reich,  Baily,  Cornu  and 
Bailie,  and  Boys  bow  the  errors  in  Cavendish's  experiment  have 
been  avoided. 

Mnncke  (61,  vol.  3,  pp.  940-70)  has  given  an  account  of  the 
experiment  and  an  admirable  criticism  of  it,  and  compares  the 
result  with  that  obtained  by  Maskelyne  and  Hufcton.  He  pro- 
posed another  method  of  using  the  torsion  balance  to  find  the 
mean  density  of  the  earth  ;  he  would  find  the  time  of  vibration 
with  the  masses  first  in  the  line  of  the  balls  and  then  in  a 
line  at  right  angles  to  that  direction.  There  would  be  no  de- 
flection to  be  measured.  We  have  seen  above  that  Brandes 
gave  the  theory  of  this  experiment.  Investigations  of  this 
nature  have  been  made  by  Reich  (83),  Eotvos  (192)  and 
Braun  (193);  for  accounts  of  which  see  the  latter  part  of  this 
volume. 

A  useful  resume  of  Cavendish's  paper  was  given  by  Schmidt 
(64,  vol.  2,  pp.  481-7).  He  formed  anew  equations  from  which 
to  derive  the  value  of  the  density  of  the  earth,  and  found  5.52 
using  Cavendish's  data. 

Another  mathematical  investigation  of  the  dynamical  prob- 
lem underlying  the  Cavendish  experiment  was  made  by  Mena- 
brea  (71,  72  and  73),  in  1840.  It  is  a  very  elaborate  analysis  of 
the  whole  problem.  He  examines  the  effect  of  the  resistance 
of  the  air  on  the  time  of  vibration,  and  also  shews  how  to  find 
the  mass  of  the  earth  supposing  that  it  is  composed  of  spheroidal 
layers  of  variable  density.  In  Baily's  memoir  (79)  is  another 
elaborate  analysis,  by  Airy,  of  the  mathematical  theory  of  the 
investigation.  It  treats  especially  of  Baily's  modification  of 
the  Cavendish  experiment  (reproduced  in  Routh's  Rigid  Dy- 
namics 1882,  pt.  1,  pp.  359-364). 

An  elementary  treatment  of  the  problem  involved  is  given  by 
Gosselin  (127),  and  from  the  formula  he  arrives  at  he  derives 
the  value  of  the  mean  density  of  the  earth  as  given  by  Caven- 
dish's experiment,  and  gets  5.69.  A  similarly  elementary  treat- 
ment by  Babinet  (132)  gives  5.5. 

An  excellent  account  of  Cavendish's  work  is  given  by  Zanotti- 
Bianco  (148-J)  and  by  Poynting  (185,  pp.  40-8)  ;  in  the  latter  is 
to  be  found  a  diagram  showing  the  closeness  of  Cavendish's 
separate  results  to  the  mean. 

106 


THE    LAWS    OF    GRAVITATION 

CAVENDISH,  son  of  Lord  Charles  Cavendish  and  a 
nephew  of  the  third  Duke  of  Devonshire,  was  born  at  Nice  in 
1731  and  died  at  London  in  1810.  He  studied  at  Cambridge, 
and  becoming  possessed,  by  the  death  of  an  uncle,  of  a  large 
fortune  he  devoted  his  life  unostentatiously  to  private  scientific 
studies.  Besides  the  investigation  on  gravitational  attraction 
here  reprinted,  he  is  remarkable  for  his  researches  in  the  field 
of  chemistry,  and  has  been  called  the  "  Newton  "  of  that  sub- 
ject. He  worked  on  the  constituents  of  the  atmosphere  and 
on  hydrogen  ;  he  made  the  first  synthesis  of  water,  by  burning 
hydrogen  in  air,  and  found  the  density  of  hydrogen  to  be  fa 
(instead  of  -fa)  of  that  of  air.  He  determined  the  ratio  of  de- 
phlogisticated  to  phlogisticated  air  to  be  about  as  1  :  4.  Caven- 
dish also  made  many  researches  of  great  importance  in  the  sub- 
ject of  electricity  ;  these  have  been  collected  and  edited  by 
Clerk-Maxwell.  Perhaps  the  most  important  of  his  electrical 
investigations  is  that  which  proved  that  electrostatic  attraction 
takes  place  according  to  the  law  of  the  inverse  square  of  the 
distance.  He  is  also  the  author  of  several  papers  on  astronomi- 
cal questions.  Most  of  his  writings  are  to  be  found  in  the 
Philosophical  Transactions  of  the  period. 

107 


HISTORICAL  ACCOUNT  OF  THE  EXPERI- 
MENTS MADE  SINCE  THE  TIME 
OF  CAVENDISH 


HISTORICAL  ACCOUNT  OF  THE  EXPERI- 
MENTS MADE  SINCE  THE  TIME 
OF  CAVENDISH 


OARLINI.  In  1821,  Carlini,  director  of  the  Brera  observatory 
at  Milan,  made  a  series  of  experiments  at  the  Hospice  on  Mt. 
Cenis  in  the  Alps,  to  determine  the  length  of  the  seconds-pend- 
ulum (55).  He  was  led  to  do  so  from  considering  that  the 
Alps  offered  a  favourable  situation  for  a  determination  of  the 
mean  density  of  the  earth,  and  that  no  pendulum  experiments 
had  been  made  there  since  the  publication  of  the  fictitious 
ones  of  Coultaud  and  Mercier  (see  p.  47).  Carlini  compared 
the  time  of  vibration  of  a  simple  pendulum,  made  after  the 
general  style  of  Borda's,  with  that  of  a  standard  clock  whose 
rate  was  noted  daily.  The  height  of  the  observing  station  was 
1943  metres  above  sea-level,  in  latitude  45°  14'  10".  The  cor- 
rected length  of  the  seconds -pendulum  reduced  to  sea -level 
was  found  to  be  993.708  mm.  Matthieu  and  Biot  had  found 
the  length  of  the  "decimal"  seconds-pendulum  at  Bordeaux, 
in  lat.  44°  50'  25",  to  be  741.6151  mm.  The  calculated  length 
at  Mt.  Cenis  would  be  741.6421  mm. ;  or,  for  the  "  sexagesimal " 
seconds-pendulum  (100000  decimal  seconds  =  86 400  sexages- 
imal seconds)  993.498  mm.  The  difference  between  this  length 
and  the  observed  length  is  .210  mm.,  which  represents  the  at- 
traction of  the  mountain  on  the  pendulum. 

The  mountain  is  composed  of  schist,  marble  and  gypsum,  of 
specific  gravities  2.81,  2.86  and  2.32  respectively.  Carlini  took 
the  average  of  all  three,  2.66,  as  the  mean  density  of  the  hill. 
Assuming  that  the  hill  was  a  segment  of  a  sphere  1  geographical 
mile  in  height  and  had  a  base  of  11  miles  in  diameter,  the  at- 
traction was  calculated  to  be  5.0203,  where  3  is  the  specific 
gravity  of  the  hill.  With  the  same  units  the  attraction  of  the 

111 


MEMOIRS    ON 

earth  is  14  394A,  where  A  is  the  mean  density  of  the  earth.* 
5.0203         .210 


We  cannot  place  very  great  confidence  in  this  result,  not  only 
on  account  of  the  fact  that  the  extreme  value  of  the  length  of 
the  seconds-pendulum  varies  from  the  mean  by  .032  mm.  and 
only  13  determinations  were  made  ;  but  especially  because  the 
size  and  density  to  be  assigned  to  the  mountain  are  largely  a 
matter  of  conjecture. 

A  resume  of  Carlini's  paper  was  given  by  Saigey  (56),  and 
by  Schell  (135).  Excellent  accounts  of  the  experiment,  with 
criticisms  of  it,  have  been  given  by  Zanotti-Bianco  (148J,  pt.  2, 
pp.  136-45),  Poynting  (185,>pp.  22-4)  and  Fresdorf  (186£,  pp. 
8-11). 

Sabine  (58  and  82,  notes,  p.  47)  remarks  that  Biot  and  Car- 
lini  had  not  properly  reduced  to  vacuo  the  observed  pendulum 
lengths,  and  states  that  the  corrected  length  of  the  seconds- 
pendulum  on  Mt.  Cenis  is  39.0992  in.  From  the  observations 
made  in  the  Formentera-Dunkirk  survey  he  finds  by  interpo- 
lation a  pendulum  length  of  39.1154  in.  for  the  latitude  of  the 
Hospice.  The  difference  between  the  observed  and  calculated 
lengths  is  .0162  in.  The  difference  calculated  from  the  inverse- 
square  law  is  .0238  in.  With  Carlini's  data  and  equations  he 
derives  4.77  for  the  value  of  A. 

Schmidt  (64,  vol.  2,  p.  480)  gives  a  concise  account  of  the 
theory  of  the  experiment,  and  remarks  that  Carlini  made  an 
error  in  determining  the  attraction  of  a  spherical  segment. 
Making  the  necessary  correction  he  finds  A  =  4.  837,  a  result 
not  far  from  that  of  the  Schehallien  experiment. 

In  1840,  Giulio  (70)  also  gave  the  true  expression  for  the  at- 
traction of  a  spherical  segment,  and  noted  that  several  other 
corrections  must  be  made  in  Carlini's  calculations  ;  the  height 
of  the  segment  is  1.05  miles,  iwstead  of  1  mile  ;  the  length  of 
the  pendulum  as  determined  by  Biot  must  be  corrected  for  an 
error  in  the  rule  used  to  find  the  length,  and  for  the  altitude  of 
Bordeaux.  Moreover,  the  reductions  to  vacuo  had  not  been  made 
properly  in  either  case.  When  all  these  corrections  had  been 
applied  to  Carlini's  results,  Giulio  found  the  value  of  A  to  be  4.95. 

*  This  signification  of  A  will  be  retained  throughout  the  rest  of  the 
volume. 

112 


THE    LAWS    OF    GRAVITATION 

Saigey  (74,  p.  155)  makes  the  observed  pendulum  length 
corrected  to  vacuo  993.756  mm.,  and  the  calculated  length 
993.617  mm.  With  these  numbers  the  value  of  A  becomes  6.15. 

Zanotti- Bianco  (14%  pt.  2,  p.  136)  mentions  that  Knopf 
(149^)  has  compared  the  value  of  gravity  as  observed  by  Car- 
lini  on  the  top  of  the  mountain  with  the  value  calculated  for 
the  same  place  from  observations  made  on  the  same  parallel  of 
latitude,  and  found  for  A,  5.08. 

AIRY,  WHEWELL  AND  SHEEPSHANKS  AT  DOLCOATH  MINE. 
In  1826,  Drobisch,  in  an  appendix  to  a  pamphlet  on  the  figure 
of  the  moon  (57),  suggested  that  experiments  be  made  on  the 
change  in  the  period  of  a  pendulum  when  carried  from  the  sur- 
face of  the  earth  to  the  bottom  of  a  mine;  he  gave  the  theory 
of  the  experiments  and  calculated  the  change  resulting  from 
certain  hypotheses.  It  is  interesting  to  recall  the  fact  that 
Bacon  proposed  the  same  investigation  two  centuries  earlier. 
(See  p.  1.) 

At  the  very  same  time,  unknown  to  Drobisch,  experiments 
of  this  nature  were  being  tried  in  England  by  Airy  and  Whew- 
ell  at  the  copper  mine  of  Dolcoath  in  Cornwall.  Their  meth- 
od was  to  swing  one  invariable  pendulum  at  the  mouth  of 
the  pit  and  compare  its  rate,  by  Rater's  method  of  coincidences, 
with  that  of  a  standard  clock,  and  at  the  same  time  perform 
the  same  operation  upon  another  pendulum  and  another  clock 
at  a  depth  of  1220  ft.  in  the  mine.  The  pendulums  were  then 
exchanged  and  the  operations  repeated.  The  greatest  difficulty 
experienced  was  that  of  comparing  the  rates  of  the  two  clocks. 
The  first  series  of  experiments  was  abruptly  stopped  on  account 
of  the  damage  received  from  fire  by  the  lower  pendulum.  A 
short  account  of  the  method  was  published  in  1827  (60),  and 
Drobisch  translated  it  for  Poggendorff's  Annalen  (59),  wherein 
he  gives  also  a  more  complete  account  of  the  theory  and  an  ap- 
plication of  his  equations  to  Airy's  observations.  Assuming 
the  mean  density  of  the  surface  layer  of  the  earth  to  be  2.587, 
the  experiments  gave  about  20  for  the  value  of  A.  Drobisch 
contends  that  the  surface  density  should  be  taken  to  be  1/52, 
considering  how  large  an  amount  of  the  surface  layer  is  water. 

Two  years  later  Airy  and  Whewell,  assisted  by  Mr.  Sheep- 
shanks and  others,  attempted  to  repeat  the  experiments  ;  but 
after  overcoming  various  anomalies  in  the  motions  of  the  pend- 
H  113 


MEMOIRS    ON 

ulums,  the  observations  were  stopped  by  a  fall  of  rock  in  the 
mine.  The  value  of  A  found  from  this  series  was  about  6.  A 
full  account  of  the  experiments  was  printed  privately  (62)  in 
1828,  and  Drobisch  translated  the  pamphlet  for  the  Annalen 
(63). 

REICH'S  FIRST  EXPERIMENT.  In  1838,  F.  Reich,  Professor 
of  Physics  in  the  Bergakademie  at  Freiberg,  published  in  book 
form  (67)  the  account  of  a  series  of  experiments  carried  on  by 
him  since  1835  to  find,  after  the  method  of  Cavendish,  the 
mean  density  of  the  earth.  The  adoption  of  the  mirror  and 
scale  method  of  measuring  deflections  seemed  to  him  to  prom- 
ise a  means  of  overcoming  many  of  the  difficulties  against  which 
Cavendish  had  contended.  The  final  observations  were  made 
in  the  year  1837. 

In  order  to  avoid  the  effects  due  to  irregularities  of  tempera- 
ture, the  apparatus  was  set  up  in  a  cellar  room  which  was  care- 
fully closed  up,  and  the  observations  made  through  a  hole  in 
the  door.  The  arm  of  the  balance  was  2.019  m.  long,  and  its 
moment  of  inertia  was  found  after  the  manner  used  by  Gauss 
for  a  magnet.  The  average  weight  of  each  of  the  balls  was 
484.213  gr.,  and  their  distance  below  the  arm  was  77  cm.  They 
were  composed  of  an  alloy  of  about  90  parts  tin,  10  parts  bis- 
muth, and  a  little  lead.  The  attracting  masses  were  of  lead 
45  kg.  in  weight  and  about  20  cm.  in  diameter,  and  hence 
much  smaller  than  those  used  ,by  Cavendish.  They  were  sus- 
pended from  pulleys  running  on  rails  parallel  to  the  arm  of  the 
balance,  and  could  be  quickly  moved  from  the  null  to  the  at- 
tracting positions.  Only  one  mass  was  in  the  attracting  posi- 
tion at  a  time,  on  account  of  the  fact  that  in  every  one  of  the 
four  attracting  positions  the  distance  from  the  mass  to  the  ball 
was  slightly  different ;  whereas  Cavendish  used  both  masses  at 
once.  The  distance  from  mass  to  ball  was  measured  at  each 
observation  by  means  of  a  telescope  moving  along  a  horizontal 
scale,  and  not  once  for  all  as  was  done  by  Cavendish. 

After  the  suspended  system  was  set  up,  Reich  found  a  con- 
tinual changing  of  the  zero-point,  which  often  lasted  for  6 
months.  In  his  final  observations  this  was  not  noticeable  be- 
cause of  the  length  of  time,  1^  years,  which  intervened  between 
the  initial  and  final  experiments.  Accordingly  the  second 
means  of  the  elongations  were  found  by  him  to  be  more  con- 

114 


THE    LAWS    OF    GRAVITATION 


stant  than  Cavendish  found  them.  The  following  table  will 
show  this,  and  also  illustrate  Reich's  method  of  finding  the 
time  of  vibration  : 


Extremes 

1st  mean 

2d  mean 

Time  of  passage 

at  74.  5 

at  75.5 

95.2, 

•• 

76.00 

IVh    27'     36".  8 

IVh   27'   30".  8 

56.8 

•• 

73.85 

74.925 

34    21  .2 

34    29  .2 

90.9 

i 

75.90 

74.875 

41     22  .4 

41     12  .4 

60.9 

•• 

74.25 

75.075 

47    59  .2 

48    10  .8 

87.6' 

Average  or  3d  mean  74.9583 
Time  of  vibration  determined  from  passage  of  74.5 

IVh  41'   22".4-IVh  27'    36". 8=13'    45". 6 
47    59  .2-          34    21  .2=13    38  .0 
Time  of  vibration  determined  from  passage  of  75.5 

IVh  41'    12".4-IVh  27'    30".8=13'    41".6 


48    10  .8- 


34    29  .2=13    41  .6 


aver.  13'   41". 6 


By  interpolation  the  time  of  a  double  vibration  across  74.9583 
is  13' 41". 708.  This  differs  somewhat  from  Cavendish's  method, 
as  will  be  seen  by  a  reference  to  page  65.  Reich  considered  it 
a  more  accurate  method  than  that  of  Cavendish,  and  remarks 
that  when  he  applied  the  hitter's  method  to  the  above  observa- 
tions he  got  results  not  very  different,  but  on  applying  his  own 
method  to  Cavendish's  observations  he  got  somewhat  different 
results.  Reich's  method  was  adopted  by  Baily  (79,  pp.  44  and 
47)  in  his  experiments. 

When  in  the  above  way  at  least  three  passages  across  the  two 
median  points  had  been  observed,  Reich  waited  until  the  arm 
was  in  its  next  extreme  position  and  seemingly  at  rest  and  then 
rapidly  moved  the  attracting  mass  to  its  new  position.  It  was 
always  so  moved  as  to  increase  the  swing.  He  assumed  that 
the  motion  was  instantaneous,  and  used  the  last  extreme  of  the 
one  series  as  the  first  of  the  next.  Baily  (79,  p.  46)  followed 
him  in  this  departure  from  the  procedure  of  Cavendish.  Cornu 
and  Bailie  (142)  have  pointed  out  that  this  method  leads  to 
error,  and  we  shall  refer  to  the  matter  again  when  describing 
the  results  obtained  by  Baily  (page  116).  From  each  such  set 
of  4  extremes  the  resting-point  and  the  time  of  vibration  are 

115 


MEMOIRS    ON 

found.  From  2  such  sets  the  deviation  could  be  found,  and 
the  mean  density  of  the  earth  calculated.  Reich  did  not,  how- 
ever, proceed  in  that  way,  but  deduced  one  value  of  A  from  all 
the  observations  of  each  day  ;  that  is,  he  took  the  average  of 
all  the  deviations  of  that  day  for  the  final  mean  deviation,  and 
the  average  of  all  the  times  of  vibration  for  the  final  mean  time 
of  vibration,  and  from  these  deduced  one  value  for  the  mean 
density  of  the  earth. 

In  applying  corrections  to  the  equations  derived  from  a  simpli- 
fied form  of  the  theory  of  the  experiment,  Reich  followed  Cav- 
endish exactly.  57  observations  were  made,  from  which  14  de- 
terminations of  the  value  of  A  were  deduced.  The  mean  of  all, 
when  corrected  for  the  centrifugal  force,  was  5. 44 ±.0233,  a  re- 
sult almost  coinciding  with  Cavendish's.  Reich  admits  at  the 
end  of  his  paper  that  there  were  certain  anomalies  in  the 
motion  of  the  beam  which  he  could  not  account  for. 

A  second  series  of  6  observations  with  iron  masses  30  kg.  in 
weight  and  20  cm.  in  diameter  gave  for  A,  5.4522,  which  proves 
that  no  disturbance  could  have  arisen  from  magnetic  action. 

Valuable  concise  accounts  of  Reich's  experiment  are  given 
by  Beaumont  (66),  Baily  (68,  69  and  79,  pp.  96-8),  Schell 
(135),  Poynting  (185,  pp.  48-50)  and  Fresdorf  (186£,  pp.  20-2). 

BAILY.  While  Reich  was  making  the  investigations  just  re- 
ferred to,  a  very  comprehensive  and  elaborate  series  of  experi- 
ments upon  almost  the  same  plan  was  being  carried  on  by  the 
English  astronomer  F.  Baily.  These  experiments  were  under- 
taken at  the  instance  of  the  Royal  Astronomical  Society,  and 
in  aid  of  them  a  grant  of  £500  was  made  by  the  British  govern- 
ment. The  results  were  published  in  1843  (79).  They  were 
carried  on  in  one  of  the  rooms  of  Baily's  residence,  a  one-story 
house  standing  detached  in  a  large  garden.  The  apparatus  was 
almost  the  counterpart  of  that  of  Cavendish,  except  that  the 
balls  were  not  suspended  from  the  balance  arm,  but  were 
screwed  directly  on  to  its  ends.  The  balance  and  its  mahogany 
case  were,  moreover,  suspended  from  the  ceiling,  and  the  at- 
tracting masses  rested  on  the  ends  of  a  plank  movable  on  a 
pillar  rising  from  the  floor.  As  a  protection  against  changes 
of  temperature  this  apparatus  was  then  surrounded  by  a  wooden 
enclosure.  The  masses  were  of  lead  rather  more  that  12  in.  in 
diameter,  weighing  380,469  Ibs.  each.  Torsion  rods  of  deal  and 

116 


THE    LAWS    OF    GRAVITATION 

of  brass,  each  about  77  in.  long,  were  employed,  and  their 
motion  was  observed  by  the  mirror  and  scale  method.  Balls  of 
different  materials  and  of  various  diameters  were  experimented 
upon  :  viz.,  1.5  in.  platinum,  2  in.  lead,  2  in.  zinc,  2  in.  glass, 
2  in.  ivory,  2.5  in.  lead,  and  2.5  in.  hollow  brass.  The  mode 
of  suspension  was  varied  greatly,  both  single  and  double  sus- 
pension wires  being  used,  and  the  material  and  distance  apart 
of  the  bifilar  wires  being  frequently  changed.  The  length  of 
the  suspending  wires  was  ordinarily  about  60  in.,  and  the  time 
of  vibration  varied  from  about  100  to  580  seconds. 

The  experiments  were  begun  in  Oct.  1838,  and  carried  on 
for  18  months,  until  about  1300  observations  had  been  made  ; 
when,  on  account  of  the  great  discordance  of  the  results,  a 
stop  was  made.  Prof.  Forbes  suggested  that  these  anomalies 
might  arise  from  radiation  of  heat,  and  advised  the  use  of  gilt 
balls  and, a  gilt  case.  These  changes  were  made,  and  the  tor- 
sion box  also  lined  with  thick  flannel.  They  turned  out  to  be 
decided  improvements,  although  some  anomalies  still  existed, 
and  it  is  evident  that  the  choice  of  a  place  for  setting  up  the 
apparatus  was  not  a  good  one. 

Baily  adopted  the  method  of  Reich  for  reducing  the  time  re- 
quired to  make  the  number  of  turning-points  requisite  for  cal- 
culating the  deviation  and  period  ;  that  is,  the  masses  were 
moved  quickly  from  one  near  position  to  the  other,  and  the  last 
turning-point  of  one  series  served  for  the  first  of  the  next. 
Three  new  turning-points  were  observed  at  each  position  of  the 
masses,  and  each  group  of  4  was  called  an  "experiment." 
2153  such  experiments  were  made  during  the  years  1841-2. 
The  time  of  vibration  was  found  for  each  experiment  after  the 
method  adopted  by  Reich.  In  deducing  the  mean  density  of 
the  earth  from  the  observations  Baily  proceeded  quite  differ- 
ently from  Reich.  There  was  always  a  slow  motion  of  the  zero- 
point,  and  Baily,  in  order  to  take  account  of  this,  combined 
the  deflections  and  periods  in  threes.  The  difference  between 
the  deflection  of  the  2d  experiment  and  the  average  of  the  1st 
and  3d  is  twice  the  mean  deviation.  The  average  of  the  period 
of  the  2d  experiment  with  the  average  of  the  1st  and  3d  is  the 
mean  period.  From  the  mean  deviation  and  mean  period  so 
found  a  value  of  A  is  deduced.  Another  was  then  found  from 
comparing  the  3d  experiment  with  the  2d  and  4th,  and  so  on. 
The  mean  of  all  the  experiments  gave  for  A,  5.6747±.0038.  Some 

117 


MEMOIRS    ON 

of  the  experiments  were  made  with  the  brass  rod  alone,  without 
any  balls,  the  mean  result  for  which  was  5. 0666 ±.0038. 

The  mathematical  analysis  of  the  problem  was  given  by 
Airy,  and  is  incorporated  in  Daily's  paper  (79,  pp.  99-111);  it 
is  also  to  be  found  in  Routh's  Rigid  Dynamics,  1882,  pt.  1, 
pp.  359-64. 

Baily  published  a  condensed  account  of  his  work  in  several 
journals  (75,  76,  77,  78  and  80).  A  careful  discussion  of  it  is 
given  by  Schell  (135)  and  by  Poynting  (185,  pp.  52-7). 

In  1842,  Saigey  (74)  wrote  a  full  account  of  all  the  experi- 
ments made  before  that  date  ;  he  gives  his  reasons  for  consider- 
ing the  pendulum  method  of  finding  A  the  least  accurate,  the 
mountain  method,  somewhat  better,  and  the  torsion  method  the 
best.  He,  finds  great  fault  with  the  work  of  Baily,  and  con- 
siders that  his  results  are  not  so  worthy  of  confidence  as  those 
of  Cavendish.  Saigey  contends  that  the  anomalies  observed 
by  Cavendish,  Reich,  and  Baily  cannot  be  accounted  for  by 
radiation  of  heat,  as  Forbes  suggested,  because  the  balance 
swings  in  an  enclosure  all  points  of  which  are  at  the  same  tem- 
perature (thus  begging  the  question);  he  confidently  remarks 
that  these  anomalies  are  caused  by  the  passage  of  air  into  or 
out  of  the  case  as  the  barometric  pressure  changes.  The  values 
of  A  found  by  Baily  increased  from  5.61  to  5.77  as  the  density 
of  the  balls  used  changed  from  21.0  to  1.9  respectively;  Saigey 
thinks  that  this  must  arise  from  an  error  in  calculating  the 
moment  of  inertia  of  the  balance  arm.  He  devises  a  graphical 
method  of  making  proper  allowance  for  this  supposed  error, 
and  deduces  as  the  final  mean  of  all  the  experiments  of  Baily  a 
value  5.52,  the  extremes  being  5.49  and  5.55. 

Saigey  made  a  new  determination  of  A  (74,  vol.  12,  p.  377), 
from  the  difference  6". 86  of  the  astronomical  and  geodetical 
latitudes  of  Evaux  as  calculated  by  Puissant.  Applying  the 
method  used  by  Hutton  for  Schehallien,  and  later  by  James 
and  Clarke  for  Arthur's  Seat,  he  found  the  ratio  of  A  to  the 
surface  density  of  France  to  be  1.7.  Assuming  the  latter  den- 
sity to  be  2.5,  the  former  becomes  4.25. 

In  1847,  Hearn  tried  to  account  for  the  anomalies  in  Baily's 
results  by  assuming  a  magnetic  action.  He  worked  out  the 
theory  (81)  of  such  action,  and  found  that  it  must  be  of  a  very 
fluctuating  nature  and  may  be  either  positive  or  negative,  and 
even  greater  in  magnitude  than  the  force  of  gravitation.  That" 

118 


THE    LAWS    OF    GRAVITATION 

such  a  magnetic  action  does  not  really  exist  is  to  be  deduced 
from  Reich's  results  with  iron  masses  (see  page  116). 

Montigny  offered  to  the  Royal  Academy  of  Belgium,  in  1852, 
a  memoir  in  which  he  attributed  the  peculiarities  in  the  be- 
haviour of  the  torsion  pendulum  in  the  experiments  of  Caven- 
dish and  of  Bciily  to  the  rotation  of  the  earth.  Schaar  (85),  to 
whom  the  memoir  was  referred  by  the  Society,  proved  that  the 
rotation  of  the  earth  could  not  produce  such  effects,  and  the 
memoir  was  not  published. 

It  was  Cornu  and  Bailie  who  first  pointed  out  (142),  in  1878, 
the  main  error  in  Baily's  method.  It  lies  in  his  taking  the  4th 
reading  of  the  turning-point  of  one  series  of  experiments  as  the 
1st  of  the  next,  as  already  explained.  They  shewed  that  the 
rotation  of  the  plank  holding  the  masses  could  not  be  per- 
formed rapidly  enough  to  get  the  masses  into  the  new  position 
before  the  arm  had  begun  its  return  journey.  They  therefore 
rejected  the  1st  of  each  series  of  4  readings,  and  calculated  A 
from  the  other  3  in  10  cases  taken  at  random  from  some  of 
Baily's  most  divergent  values,  and  found  5.615  instead  of  5.713. 
Reducing  Baily's  final  value  in  the  same  proportion  they  got 
5.55. 

A  curious  relation  between  density  and  temperature  as  pre- 
sented in  Baily's  determinations  was  pointed  out  by  Hicks 
(166),  in  1886.  The  mean  density  seems  to  fall  with  rise  of 
temperature.  The  most  probable  explanation  of  this  is  given 
by  Poynting  (185,  p.  56),  who  remarks  that  the  experiments 
with  the  light  balls  happened  to  be  made  in  winter,  and  those 
with  the  heavy  balls  in  summer.  Hicks  also  refers  to  several 
slight  corrections  to  be  made  in  Airy's  discussion  of  the  theory 
—viz.,  for  the  air  displaced  by  the  attracting  masses,  for  the 
inertia  of  the  air  in  which  the  balls  move,  and  for  expansion 
with  change  of  temperature. 

REICH'S  SECOND  EXPERIMENT.  Ten  years  after  the  appear- 
ance of  Baily's  memoir,  Reich  published  (83)  an  account  of 
some  farther  experiments  with  his  apparatus.  In  the  begin- 
ning of  his  paper  he  pointed  out  that  Baily's  method  of  com- 
bining the  results  of  the  separate  experiments  was  better  than 
that  used  by  himself.  He  proceeded  to  calculate  the  results 
of  his  first  experiments  by  Baily's  method  and  found  for  A  the 
value  5.49±.020. 

119 


MEMOIRS    ON 

Being  impressed  with  the  anomalies  in  Baily's  observations, 
and  especially  with  the  variation  of  the  final  results  with  the 
density  of  the  balls,  Reich  determined  to  repeat  his  experi- 
ments. His  apparatus  was  set  up  this  time  in  a  second-story 
room,  and  Baily's  devices  were  employed  in  order  to  avoid  the 
effects  of  temperature  changes.  The  only  important  change  in 
the  arrangement  of  the  apparatus  was  in  the  placing  of  the  large 
mass.  It  was  now  set  in  one  of  four  depressions  90°  apart  in  a 
circular  table  revolving  under  the  balance  about  a  vertical  axis 
passing  through  the  centre  of  one  of  the  balls ;  thus  no  correc- 
tion was  necessary  for  the  attraction  of  the  table  and  its  sup- 
ports upon  the  ball.  The  balls  and  masses  were  those  used  in 
the  first  experiment.  Three  series  of  experiments  were  made 
during  the  years  1847-50,  one  with  a  suspending  wire  of  thin 
copper,  one  with  thick  copper,  and  one  with  a  bifilar  iron  sus- 
pension. The  final  mean  density  of  the  earth  was  found  to  be 
5.5832±.0149. 

In  order  to  make  a  test  of  Hearn's  explanation  (see  page  118) 
of  the  peculiarities  in  Baily's  results,  Reicli  made  some  further 
experiments.  He  kept  the  North  pole  of  a  strong  magnet  near 
the  attracting  lead  mass  for  a  whole  day,  and  then  suddenly 
rotated  the  mass  through  180°  about  a  vertical  axis;  but  no 
effect  was  evident.  Hence  variations  in  the  result  are  not  due 
to  the  magnetizing  of  the  masses  by  the  earth,  or  similar 
causes.  He  then  took  off  the  tin  balls  and  substituted  success- 
ively balls  of  bismuth  and  of  iron.  The  values  of  A  were  re- 
spectively 5.5233  and  5.6887  ;  the  largeness  of  the  latter  denotes 
possibly  a  diamagnetic  action  of  the  lead  mass ;  but  it  shews 
that  under  the  original  circumstances  no  measurable  effect 
could  have  arisen  from  magnetic  action. 

Prof.  Forbes  had  suggested  *  to  Reich  that  A  could  be  found 
from  the  period  of  the  balance  only,  by  noting  the  variation  of 
the  time  of  vibration  with  the  position  of  the  attracting  masses. 
'Reich  made  some  experiments  of  this  nature  by  placing  two 
lead  masses  diametrically  opposite  to  each  other,  first  so  that 
the  line  joining  them  was  perpendicular  to  the  vertical  plane 
through  the  torsion  arm,  and  next  was  in  the  plane.  This 
caused  no  deviation,  but  only  a  change  in  the  time  of  swing  of 


*  We  have  seen  (page  106)  that  this  method  was  suggested  earlier,  in 
Gehler's  PhysikaliscJie  Worterbuch,  and  the  equations  given  by  Brandes  (42). 

120 


THE    LAWS    OF    GRAVITATION 

the  balance.     The  value  of  A  found  in  this  way  was  6.25,  but 
the  apparatus  was  not  well  devised  for  the  work. 

Several  abstracts  of  Reich's  paper  are  to  be  found  (84,  86, 
87  and  185,  pp.  50-2). 

AIRY'S  HARTON  COLLIERY  EXPERIMENT.  We  have  already 
referred  to  Airy's  experiments  in  the  Dolcoath  mine  in  1826-8. 
In  1854,  he  again  undertook  to  carry  out  investigations  (100) 
along  the  same  lines,  the  introduction  of  the  telegraph  having 
made  easy  the  comparison  of  the  clocks  at  the  top  and  bottom 
of  the  mine.  He  selected  the  Harton  Colliery,  near  South 
Shields,  for  the  experiments,  which  were  carried  out  by  six  ex- 
perienced assistants  of  whom  Mr.  Dunkin  was  the  chief.  The 
two  stations  were  vertically  above  each  other  and  1256  ft.  apart. 
The  apparatus  was  the  best  obtainable,  and  special  precautions 
were  taken  in  order  that  the  pendulum  supports  might  be  rigid. 

Simultaneous  observations  of  the  two  pendulums  were  kept 
up  night  and  day  for  a  week  ;  then  the  pendulums  were  ex- 
changed and  observations  taken  for  another  week.  Two  more 
exchanges  were  made,  but  the  observations  for  them  both  were 
made  in  one  week.  Each  pendulum  had  six  swings  of  nearly 
4  hours  each  on  every  day  of  observation,  and  between  success- 
ive swings  the  clock  rates  were  compared  by  telegraphic  signals 
given  every  15  seconds  by  a  journeyman  clock. 

The  corrections  and  reductions  were  carried  out  by  Airy  in  a 
very  elaborate  manner.  The  results  of  the  1st  and  3d  series 
agree  very  closely,  as  do  those  of  the  2d  and  4th,  showing  that 
the  pendulums  had  undergone  no  sensible  change.  By  com- 
paring the  mean  of  the  1st  and  3d  series  with  the  mean  of  the 
2d  and  4th,  the  ratio  of  the  pendulum  rates  at  the  upper  and 
lower  stations  is  obtained  independently  of  the  pendulums  em- 
ployed. The  final  result  gave  gravity  at  the  lower  station 
greater  than  gravity  at  the  upper  by  Trrrs^1  part,  with  an  un- 
certainty of  g+oth  Part  °f  ^ie  incl'ease  ;  or  the  acceleration  of 
the  seconds-pendulum  below  is  2". 24  per  day,  with  an  uncer- 
tainty of  less  than  0".01. 

In  order  to  calculate  what  this  difference  should  be,  suppose 
the  earth  to  be  a  sphere  of  radius  r  and  mean  density  A,  sur- 
rounded by  a  spherical  shell  of  thickness  h  and  density  S,  then 

,  gravity  below  2h      37/3  , 

a  simple  analysis  shews  that  —  —  —  1-| (corn- 

gravity  above  r       rA 

121 


MEMOIRS    ON 

pare  p.  31).  Airy  gives  a  discussion  of  the  effect  of  surface  ir- 
regularities ;  it  is  shewn  that,  supposing  the  surface  of  the  earth 
near  the  mine  to  have  no  irregularities,  the  effect  of  those  at  dis- 
tant parts  of  the  earth  may  be  neglected.  He  also  assumes  that 
there  is  no  sudden  change  of  density  just  under  the  mine.  He 
proves  that  the  effect  of  a  plane  of  3  miles  in  radius  and  of  the 
thickness  of  the  sh'jll  is  ff  of  that  of  the  whole  shell,  so  that 
only  the  neighbouring  country  need  be  surveyed.  Since  the 
upper  station  is  only  74  ft.  above  high  water,  it  will  be  sufficient 
to  assume  that  any  excess  or  defect  of  matter  exists  actually  on 
the  surface.  A  careful  survey  of  the  environs  of  the  mine  was 
made,  and  allowance  made  for  each  elevation  and  depression. 
The  general  result  is  that  the  attraction  of  the  regular  shell  of 

matter  is  to  be  diminished  by  about  ^-th  part  :  •  -  — 

gravity  above 

=  1.00012032-.  00017984  x-.      Now  from   the  pendulum  ex- 


periments Airy  found  =  1.00005185  ±.  00000019  ; 

gravity  above 

hence  ^=2.  6266  ±.0073.     Prof.  W.  H.  Miller  found  the  aver- 
o 

age  density  of  the  rocks  in  the  mine  to  be  2.50;  hence  Ar^6.566 
±.0182. 

Airy  had  intended  that  the  temperatures  at  the  two  stations 
should  be  the  same,  but  the  temperature  of  the  lower  station 
was  7°.  13  F.  higher  than  that  of  the  upper.  In  a  supplement- 
ary paper  (101)  Airy  makes  a  correction  for  this  temperature 
difference  in  two  distinct  ways,  giving  for  the  corrected  A,  6.809 
and  6.623  respectively.  In  this  paper  Stokes  (102)  investigates 
the  effect  of  the  earth's  rotation  and  ellipticity  in  modifying 
the  results  of  the  Harton  experiments.  It  was  found  to  be 
small,  changing  A  from  6.566  to  6.565. 

Airy  published  several  preliminary  notices  of  his  work  (88,  89 
and  122),  abstracts  of  which  appeared  in  several  journals  (90, 
91,  92,  98  and  111).  Valuable  resumes  of  the  main  paper  are 
also  to  be  found  (105,  107,  109,  112  and  119). 

Haughton  (106,  110,  113  and  116)  gave  a  rough  but  simple 
method  of  deducing  A  from  Airy's  figures,  and  arrived  at  5.48 
as  the  value  of  A.  Knopf  (149|)  has  severely  criticized  this 
calculation.  Another  simple  formula  for  the  same  purpose 
was  given  by  an  anonymous  writer  (114).  On  the  effect  of 

122 


THE    LAWS    OF    GRAVITATION 

great  changes  in  density  below  the  under  station  one  should  read 
the  paper  by  Jacob  (118  and  121)  already  referred  to.  Schef- 
fler  (134)  published  in  1865,  though  it  is  dated  1856,  the  pro- 
posal of  an  experiment  similar  to  Airy's,  but  made  no  reference 
to  any  earlier  proposals  of  the  same  kind.  Folie  (136)  calcul- 
ated, in  1872,  the  attraction  at  the  two  stations  in  a  manner 
different  from  Airy's,  by  considering  the  shell  as  made  up  of  2 
parts.  Using  Airy's  data  he  arrived  at  6.439  as  the  value  of  A. 
Valuable  summaries  and  criticisms  of  Airy's  work  are  given 
by  Schell  (135),  Zanotti-Bianco  (148J,  pt.  2,  pp.  146-60),  Poynt- 
ing  (185,  pp.  24-9)  and  Fresdorf  (186$,  pp.  13-7). 

JAMES  AND  CLARKE.  As  a  result  of  the  calculations  made 
from  the  observations  taken  for  the  Ordnance  Survey  of  Great 
Britain  and  Ireland  (104,  117,  120,  124,  125  and  126)  by  Lt. 
Col.  James,  it  was  found  that  the  plumb-line  was  considerably 
deflected  at  several  of  the  principal  trigonometrical  stations. 
It  was  evident  from  the  nature  of  the  ground  at  the  places 
under  consideration,  that  this  deflection  was  due  to  irregulari- 
ties of  the  surface.  In  order  to  study  this  action  more  care- 
fully James  decided  to  have  the  Schehallien  experiment  re- 
peated at  Arthur's  Seat,  near  Edinburgh  (103  and  125,  pp.  572- 
624).  The  observations  were  made  during  Sept.  and  Oct., 
1855,  with  Airy's  zenith -sector,  on  the  summit  of  Arthur's 
Seat  (A),  and  at  points  near  the  meridian  on  the  north  (N) 
and  south  (S)  of  that  mountain,  at  about  one-third  of  its  al- 
titude above  the  surrounding  country.  After  corrections  had 
been  applied,  the  results  were  as  follows  : 

Station  Astronomical  lat.EEA  Geodetical  lat.=G  A— G 

8  55°  56'  26".69  55°  56'  24".25  2".44 

A  56  43  .69  56  38  .44  5  .25 

N  57     9  .22  57     2  .71  6  .51 

It  will  be  noticed  that  even  on  the  summit  of  the  hill  there 
is  an  attraction  of  more  than  5"  toward  the  south,  which  can 
not  be  due  to  the  hill.  Similarly,  to  the  south  of  the  hill  the 
attraction  is  not  toward  the  north  as  we  might  expect.  It  is 
evident  that  there  is  present  some  other  attracting  force,  be- 
sides that  of  Arthur's  Seat,  which  appears  to  produce  a  general 
deflection  of  5"  toward  the  south. 

123 


MEMOIRS    ON 

Capt.  Clarke,  who  made  all  the  calculations,  in  order  to  find 
the  attraction  according  to  Newton's  law,  used  a  modification  of 
the  method  of  Hutton.  He  took  account  of  all  the  surface 
irregularities  within  a  radius  of  about  24000  ft.  The  resulting 
value  for  the  ratio  of  the  density  of  the  rock  composing  the  hill 
to  that  of  the  whole  earth  was  .5173  ±.0053.  James  investi- 
gated the  density  of  the  rocks  of  Arthur's  Seat  and  found  it 
to  be  on  the  average  2.75.  This  gives  for  A  the  value  5.316 
±.054. 

In  order  to  see  whether  the  general  deflection  of  5"  could  be 
accounted  for  by  the  presence  of  the  hollow  of  the  River  Forth 
to  the  north  and  the  high  land  of  the  Pentland  Hills  to  the 
south,  Clarke  extended  the  calculated  attraction  to  the  borders 
of  Edinburghshire,  some  13  miles  away.  He  was  able  in  this 
way  to  account  fora  general  deflection  of  2".52,  and  he  thought 
that  by  carrying  the  calculations  to  Peeblesshire  the  whole  5" 
might  be  accounted  for. 

Several  abstracts  of  the  original  paper  have  been  published 
(108,  115  and  123).  Poynting  (185,  pp.  19-22)  has  given  a 
valuable  criticism  of  the  work. 

In  connection  with  this  investigation  might  be  mentioned 
the  various  writings  on  the  subject  of  local  attractions.  Any 
one  wishing  to  become  acquainted  with  this  subject  should  read 
Airy's  account  of  his  "  flotation  theory  "  (94  and  97),  Faye's 
account  of  his  "compensation  theory"  (130,  146J  and  147), 
Pratt's  papers  (93,  96  and  99),  Saigey  (74),  Struve  (129),  Pech- 
mann  (131),  the  treatises  of  Pratt  (133),  Clarke  (149)  and  Hel- 
mert  (148,  vol.  2).  Many  other  references  to  papers  by  these 
men  as  well  as  by  Schubert,  Peters,  Keller,  Bauernfeind  and 
others  are  to  be  found  in  the  Roy.  Soc.  Cat.  of  Scientific  papers 
and  in  Gore's  "  A  Bibliography  of  Geodesy"  (174).  See  also 
note  on  page  31  and  remarks  on  page  56.  We  might  here  re- 
call the  determination  of  A  by  Saigey  from  local  attraction  (see 
page  118).  Pechmann  (131)  in  the  same  way  found  in  the  Tyrol, 
in  1864,  two  different  values  for  A,  6.1311  ±.1557  and  6.352± 
.726,  having  assumed  the  density  of  the  earth's  crust  to  be 
2.75.  We  shall  refer  later  on  to  the  determinations  of  Men- 
denhall  and  Berget. 


AND  BAILLE.     In  1873,  Cornu  and  Bailie  published 
a  short  paper  (137)  stating  that  they  had  undertaken  to  repeat 

124 


THE   LAWS    OF   GRAVITATION 

the  Cavendish  experiment  under  conditions  as  different  as  pos- 
sible from  those  previously  employed.  They  began  by  making 
a  thorough  study  of  the  torsion-balance  in  order  to  learn  under 
what  conditions  it  would  have  the  greatest  precision  and  sensi- 
tiveness. They  found  among  other  things  that  the  resistance  of 
the  air  was  proportional  to  the  velocity  (141, 142,  143  and  157). 

The  apparatus  was  set  up  in  the  cellar  of  the  iScole  Polytech- 
nique.  The  arm  of  the  balance  was  a  small  aluminium  tube 
50  cm.  long,  carrying  on  each  end  a  copper  ball  109  gr.  in 
weight.  The  suspension  wire  was  of  annealed  silver  4.15  m. 
long,  and  the  time  of  vibration  of  the  system  6'  38".  The  at- 
tracting mass  was  mercury  which  could  be  aspirated  from  one 
spherical  iron  vessel  on  one  side  of  one  of  the  copper  balls  to 
another  vessel  similarly  situated  on  the  other  side  of  the  ball. 
This  method  got  rid  of  the  disturbances  arising  from  the  move- 
ment of  the  lead  masses  in  the  Cavendish  form  of  the  experi- 
ment. The  iron  vessel  was  12  cm.  in  diameter  and  the  mer- 
cury weighed  12  kg.  Another  great  improvement  was  the 
reduction  of  the  dimensions  of  the  apparatus  to  J  of  that  used 
by  Cavendish,  Reich  and  Baily,  the  time  of  oscillation  and  the 
sensitiveness  remaining  the  same.  The  motion  of  the  arm  was 
registered  electrically. 

Two  series  of  observations  were  made ;  one  in  the  summer 
of  1872  gave  A  =  5. 56,  and  the  other  in  the  following  winter 
5.50.  The  difference  was  explained  by  a  flexure  of  the  torsion- 
rod,  and  the  former  result  was  considered  the  better. 

In  a  later  report  (142)  they  refer  to  some  changes  made  in 
their  apparatus ;  they  increased  the  force  of  attraction  by  using 
4  iron  receivers,  2  on  each  side  of  each  copper  ball,  and  they 
reduced  the  distance  between  the  attracting  bodies  in  the  ratio 
of  -v/2  to  1.  The  time  of  vibration,  408",  remained  the  same 
within  a  few  tenths  of  a  second  for  more  than  a  year.  The  new 
value  of  A  was  5.56. 

We  have  already  referred  (page  119)  to  the  fact  that  Cornu 
and  Bailie  found  out  the  error  in  the  Baily  experiments. 

A  final  account  of  these  experiments  has  not  yet  been  pub- 
lished. Abstracts  of  the  papers  cited  are  given  by  Poynting 
(185,  pp.  57-8)  and  by  several  journals  (138  and  139). 

JOLLY.  In  1878,  von  Jolly  of  Munich  published  an  account 
(144  and  145)  of  the  results  of  his  study  of  the  beam  balance 

125 


MEMOIRS    ON 

as  an  instrument  for  measuring  gravitational  attractions.  He 
discussed  the  sources  of  error  in  the  balance  readings  and 
methods  of  eliminating  them.  The  variations  due  to  tempera- 
ture effects  are  very  difficult  to  avoid,  but  by  working  in  the 
mornings  only,  and  by  covering  the  balance  case  with  another 
lined  inside  and  out  with  silver  paper,  it  was  found  to  be  pos- 
sible to  get  quite  concordant  results. 

Jolly  applied  the  balance  to  test  the  Newtonian  law  of  the 
distance.  Two  extra  scale  pans  were  suspended  by  wires  from 
the  ordinary  scale  pans  of  the  balance  and  5.29  m.  below  them. 
The  wires  and  lower  scale  pans  were  enclosed  to  prevent  oscilla- 
tions from  air  currents.  Two  kilogramme  masses  of  pplished 
nickel-plated  brass  were  balanced  against  each  other,  first  both 
in  the  upper  scale  pans,  and  then  one  in  the  upper  and  the 
other  in  the  lower  pan,  in  each  case  double  weighings  being  made 
after  the  manner  of  Gauss.  The  motion  of  the  beam  was  noted 
by  the  mirror  and  scale  method,  the  mirror  being  fixed  at  the 
middle  of  the  beam  and  perpendicular  to  its  length.  If  r  is  the 
radius  of  the  earth  at  sea-level,  and  h  a  height  above  it,  then  a 

mass  Qi  at  sea-level  weighs  Q2  at  h,  where  Q2=Qj  (l—  - )  ap- 

,Q2     1  000  000-1.5099 
proximately.   Jolly  found  by  experiment  ^-= — , 

V^j 

Q2     1  000  000-1.662 

whereas  the  equation  gives  ^= TT^TTAA; •     ^ne  differ- 

Vi 

ence,  .152  mg.,  Jolly  thought,  was  due  to  local  attractions.  He 
proposed  to  repeat  the  experiment  at  the  top  of  a  high  tower, 
and  at  the  same  time  to  find  the  mass  of  the  earth  by  noting 
the  change  in  weight  of  one  of  the  masses  in  the  balance  when 
a  large  lead  ball  was  brought  beneath  it. 

The  results  of  these  experiments  (153  and  154)  were  published 
in  1881.  The  distance  between  the  scale  pans  was  now  21.005 
m.  The  arm  of  the  balance  was  60  cm.  long,  and  the  maxi- 
mum load  5  kg.  Four  hollow  glass  spheres  of  the  same  size 
were  made  and  in  each  of  two  5  kg.  of  mercury  were  put, 
and  all  were  sealed  up.  Each  scale  pan  had  always  one  sphere 
in  it,  and  thus  air  corrections  were  avoided.  An  observation 
was  made  as  follows  :  first  the  mercury-filled  spheres  were  bal- 
anced in  the  upper  pans,  and  then  one  in  the  upper  pan  was 
balanced  against  the  other  in  the  lower.  The  change  in  weight 

126 


THE    LAWS    OF    GRAVITATION 

observed  was  31.686  mg.;  whereas  the  change  as  calculated 
from  the  formula  should  have  been  33.059*  mg.  The  differ- 
ence is  in  the  same  direction  as  in  the  earlier  experiment. 

A  sphere  of  radius  .4975  m.  and  weight  5775.2  kg.  was  then 
built  up  out  of  lead  bars  under  the  lower  scale  pan  which 
received  the  mercury-filled  globe.  The  distance  from  the 
centre  of  this  sphere  to  that  of  the  globe  was  then  .5686  m. 
The  attraction  of  the  sphere  for  the  mercury-filled  globe  when 
in  the  upper  pan  was  neglected. 

Observations  were  made  exactly  as  before,  and  the  change  in 
weight  was  32.275  mg.  The  increase  in  weight  due  to  the 
presence  of  the  lead  is  therefore  .589  mg.  Knowing  the  den- 
sity of  the  lead  to  be  11.186,  a  simple  calculation  gives  for  the 
mean  density  of  the  earth  5. 692 ±.068. 

An  account  of  these  experiments  is  given  by  Helmert  (148, 
vol.  2,  pp.  380-2),  Zanotti- Bianco  (148-|,  vol.  2,  pp.  175-82), 
Wallentin  (154J),  Keller  (167),  Poynting  (185,  pp.  61-4)  and 
Fresdorf  (186J,  pp.  23-5). 

MENDENHALL.  In  1880,  Prof.  T.  C.  Mendenhall  described 
(150)  a  method  of  finding  the  period  of  a  pendulum  such  that 
a  determination  required  20  or  30  minutes  only.  At  the  begin- 
ning and  end  of  this  time  the  pendulum  throws  a  light  trip- 
hammer of  wire  which  breaks  a  circuit  and  makes  a  record  on 
a  chronograph  on  which  a  break-circuit  clock  is  also  marking. 
The  advantage  of  such  an  arrangement,  in  addition  to  the 
short  time  required,  is  that  the  arc  of  vibration  may  be  small 
and  will  change  very  little.  Mendenhall  expressed  a  deter- 
mination to  find  the  variation  of  the  acceleration  due  to  gravity 
on  going  from  Tokio  to  the  top  of  Mount  Fujiyama. 

A  year  later  the  results  of  these  experiments  were  published 
(151),  having  been  made  in  Aug.,  1880.  An  invariable  pend- 
ulum was  used,  made  from  a  Kater's  pendulum  by  removing 
one  ball  and  knife-edge.  Its  period  at  Tokio  (barometer  30  in. 
and  temperature  23°. 5  C.)  was  .999834  sec.  On  the  top  of 
Fujiyama  the  barometer  stood  nearly  stationary  at  19.5  in. 
during  the  observations,  and  the  thermometer  at  8°. 5.  After 
approximate  corrections  were  made  for  buoyancy,  the  time, 
reduced  to  Tokio  conditions,  was  1.000336  sec.  Assuming  g  at 

*  According  to  Helmert  this  should  be  33.108  and  according  to  Zanotti- 
Bianco  33.053. 

127 


MEMOIRS    ON 

Tokio  to  be  9.7984,  as  he  had  found  in  the  previous  year,  it  fol- 
lows that  at  the  summit  of  Fujiyama  it  is  9.7886. 

No  exact  triangulation  of  the  region  had  been  made,  but 
Mendenhall  assumed  Fujiyama  to  be  a  cone  2.35  miles  high 
standing  on  a  plain  of  considerable  extent.  The  angle  of  the 
cone  was  measured  from  photographs  and  found  to  be  138°. 
Fujiyama  is  an  extinct  volcano,  said  to  have  been  made  in  a 
single  night,  and  hence  its  composition  ought  to  be  homogene- 
ous. Its  average  density  was  taken  as  2.12,  but  no  great  re- 
liance can  be  placed  on  this  number.  Corrected  for  the  differ- 
ence in  latitude,  19',  between  Tokio  and  Fujiyama,  the  time  at 
its  base,  supposing  the  hill  taken  away,  would  be  .999847  sec. 
The  density  of  the  earth,  calculated  from  these  data  after  the 
manner  of  Carlini,  was  found  to  be  5.77. 

Fresdorf  (186^,  pp.  11-13)  describes  fully  the  experiments 
and -points  out  an  error  in  Mendenhall's  calculations  ;  the  cor- 
rected value  for  A  is  5.667.  Poynting  (185,  pp.  39-40)  gives 
-an  abstract  of  the  papers  referred  to. 

STERNECK.  Major  von  Sterneck  has  made  several  investiga- 
tions of  the  variation  of  gravity  beneath  the  earth's  surface. 
The  earliest  experiments  (155)  were  made,  in  1882,  in  the 
Adalbert  shaft  of  the  silver  mine  at  Pribram  in  Bohemia.  The 
method  employed  was  to  carry  an  invariable  half-second  pend- 
ulum and  a  comparison  clock  from  one  station  to  another,  and 
find  the  period  by  the  method  of  coincidences,  the  clock  being 
compared  with  a  standard  clock  by  carrying  a  pocket  chron- 
ometer from  one  to  the  other.  The  pendulum,  of  brass,  was 
a  rod  24  cm.  in  length  carrying  a  lens-shaped  bob  weighing  1 
kg.  The  knife  was  of  steel  whose  edge  was  so  cut  away  that 
it  rested  on  a  glass  plate  on  two  points  only.  The  apparatus 
was  always  enclosed  in  a  glass  case  to  prevent  air  currents. 
The  3  stations  at  which  observations  were  made  were  at  the 
surface,  516.0  m.  and  972.5  m.  below  the  surface  respectively. 
The  respective  periods  at  these  stations  were  .5008550,  .5008410 
and  .5008415  seconds,  and  the  resulting  values  of  A,  found 
from  Airy's  formula,  were  6.28  and  5.01,  the  density  of  the 
surface  layer  being  taken  as  2.75.  It  will  be  noticed  that  the 
values  of  g  at  the  two  underground  stations  are  practically  the 
same,  and  the  results  are  unsatisfactory. 

A  year  later  (156)  von  Sterneck  repeated  his  experiments  at 

128 


THE    LAWS    OF    GRAVITATION 

the  same  stations  and  at  two  additional  ones.  In  order  that 
his  observations  might  be  independent  of  the  rates  of  the  clocks 
used  in  finding  the  periods,  Sterneck  introduced  an  important 
modification  of  the  method  adopted  by  Airy  and  by  himself  in 
his  earlier  investigations.  He  made  another  pendulum  similar 
to  the  one  described  above  ;  one  of  these  was  always  at  the 
surface  station  and  the  other  at  one  of  the  underground  sta- 
tions, and  their  relative  periods  were  compared  by  means  of 
electric  signals  sent  simultaneously  from  a  single  clock.  This 
clock  kept  a  circuit  closed  for  half  a  second  every  other  half 
second  and  operated  a  relay  with  a  strong  current  at  each  sta- 
tion. The  passage  of  the  "tail"  of  the  pendulum  in  front  of 
a  scale  was  observed  by  means  of  a  telescope,  in  the  focal  plane 
of  which  was  a  shutter  moved  by  the  relay  current  every  half 
second,  and  at  those  instants  only  was  the  picture  of  the  tail 
of  the  pendulum  allowed  to  pass  to  the  eye  through  the  tele- 
scope. The  time  of  a  coincidence  was  when  at  one  of  these 
flashes  the  tail  appeared  exactly  at  the  middle  of  the  scale  ;  the 
time  between  two  successive  coincidences  determines  the  period 
of  the  pendulum.  The  observer  at  each  of  the  two  stations  is 
thus  finding  the  period  of  his  pendulum  in  terms  of  exactly  the 
same  unit  of  time.  When  the  observations  were  corrected,  it 
was  found  that  the  period  at  the  highest  underground  station 
was  less  than  that  at  the  next  lower  station,  and  the  determin- 
ation at  the  former  station  was  consequently  not  used.  The 
values  of  A  as  determined  from  observations  at  the  other  sta- 
tions were  5.71,  5.81  and  5.80,  with  a  mean  of  5.77.  Helmert 
(148,  vol.  2,  p.  499)  has  made  a  recalculation  and  finds  that 
these  numbers  should  be  5.54,  5.71,  5.80  and  5.71  respectively. 
Von  Sterneck  used  his  results  at  the  surface  and  at  these 
underground  stations  to  express  g  as  a  function  of  the  depth. 
Calling  the  value  of  g  at  the  surface  unity,  and  measuring  r 
from  the  centre  of  the  earth  and  calling  it  equal  to  unity  at 
the  surface,  he  deduced  the  following  expression  for  the  value 
of  g  at  any  depth  : 

#  =  2.6950  r-1.8087  r2+.1182  r\ 

This  would  make  g  a  maximum,  1.06,  at  r  =  .78.  The  density 
would  be  expressed  by  the  formula  d= 15. 13  6— 12.512  r,  giving 
15.136  for  its  value  at  the  centre  of  the  earth,  and  2.624  at  the 
surface.  These  relations  are  at  least  suggestive  if  not  con- 
vincing. 

i  129 


MEMOIRS    ON 

During  the  year  1883  von  Sterneck  used  the  same  method 
and  apparatus  to  determine  the  variation  in  gravity  for  3  sta- 
tions above  the  earth's  surface  at  Kronstadt.  He  found  (158) 
gravity  greater  at  a  higher  point  (Schlossberg)  than  at  a  lower 
(Zwinger),  and  proved  that  neither  the  formula  of  Young  (see 
page  31)  nor  that  of  Faye  and  Ferrel  for  the  reduction  to  sea- 
level  gave  satisfactory  results. 

Twice  in  this  year  Sterneck  made  investigations  at  Krusna 
hora  in  Bohemia.  Here  there  was  a  mine  with  a  horizontal 
gallery  1000  m.  long,  and  he  wished  to  find  the  effect  of  the 
overlying  sheet  of  earth  upon  the  value  of  gravity  at  various 
points  in  the  gallery.  The  same  apparatus  was  used  after  some 
improvements  had  been  made.  Observations  were  taken  at  the 
mine  mouth  and  at  points  390  and  780  m.  from  the  mouth,  and 
62  and  100  m.  respectively  below  the  surface  of  the  ground. 
The  results  shewed  that  gravity  in  the  plateau  increased  with 
the  depth  of  the  super-incumbent  layer  by  the  half  of  the 
amount  by  which  it  would  have  changed  in  free  space  when 
the  distance  from  the  centre  of  the  earth  was  changed  by  the 
same  amount.  Observations  were  made  at  4  stations  above 
ground  also  at  different  elevations,  and  it  was  found  that  the 
Faye-Ferrel  rule  accounted  for  the  differences  between  them 
much  better  than  did  the  Bouguer-Young  rule. 

Further  experiments  (164)  were  made,  in  1884,  at  Saghegy 
in  Hungary,  and  elsewhere,  with  results  similar  to  those  de- 
scribed above.  An  important  improvement  was  made  in  the 
method  of  observing  the  coincidences.  They  were  now  ob- 
served by  the  reflections  of  an  electric  spark  from  two  mirrors, 
one  fixed  on  the  pendulum  stand,  and  the  other  attached  to 
the  pendulum  and  when  at  rest  parallel  to  the  first.  The  spark 
was  made  by  the  relay  circuit  every  half  second. 

In  1885,  Sterneck  made  a  series  of  observations  (165)  at  the 
mouth  and  at  4  underground  stations  in  the  Himmelfahrt- 
Fundgrube  silver  mine  at  Freiberg  in  Saxony.  He  was  led  to 
do  so  by  the  publication  of  the  results  of  some  pendulum  meas- 
urements made  there,  in  1871,  by  Dr.  C.  Bruhns,  who  had  found 
that  gravity  decreased  with  the  depth.  Using  Airy's  formula, 
von  Sterneck  found  the  following  values  for  A  at  the  4  under- 
ground stations  in  the  order  of  their  depth  :  5.66,  6.66,  7.15 
and  7.60,  the  density  of  the  mine  strata  being  2.69.  These  re- 
sults indicate  an  abnormal  increase  of  gravity  with  depth.  Von 

130 


THE    LAWS    OF    GRAVITATION 

Sterneck  noticed  that  in  these  experiments,  as  well  as  in  those 
made  at  Pribram,  the  increase  in  gravity  is  nearly  proportional 
to  the  increase  in  temperature.  But  although  Hicks  (166),  as 
we  have  seen  (page  119),  discovered  a  connection  between  the 
values  of  A  and  the  temperatures  in  Baily's  experiments,  and 
Cornu  and  Bailie  (page  125)  got  a  larger  result  for  A  in  summer 
than  in  winter,  we  have  no  reason  for  looking  upon  the  varia- 
tions in  temperature  as  an  explanation  of  the  anomalies  under 
consideration.  An  interesting  criticism  of  von  Sterneck's  work 
is  given  by  Poynting  (185,  pp.  29-39).  Short  accounts  of  it 
are  given  by  Presdorf  (186J,  pp.  17-9)  and  Giinther  (196 ^  vol. 
1,  p.  189). 

WILSING.  In  1887,  J.  Wilsing  (170)  made  at  Potsdam  a  de- 
termination of  the  mean  density  of  the  earth  by  means  of  an 
instrument  which  is  called  the  pendulum  balance,  and  is  the 
common  beam  balance  turned  through  90°.  It  is  practically  a 
pendulum  made  of  a  rod  with  balls  at  each  end  and  a  knife- 
edge  placed  just  above  the  centre  of  gravity.  The  instrument 
used  by  Wilsing  consisted  of  a  drawn  brass  tube  1  m.  long, 
4.15  cm.  in  diameter  and  .16  cm.  thick,  strengthened  near  the 
middle  where  the  knife-edge  is  affixed.  The  knife-edge  and 
the  bed  on  which  it  rested  were  of  agate,  and  6  cm.  long.  To 
the  ends  were  screwed  the  balls  of  brass  weighing  540  gr.  each, 
and  on  the  upper  ball  was  a  pin  carrying  discs  which  were  used 
for  finding  the  moment  of  inertia  and  the  position  of  the  cen- 
tre of  gravity  of  the  pendulum.  Its  motion  was  observed  by 
the  telescope  and  scale  method,  a  mirror  being  attached  to  the 
side  of  the  pendulum  parallel  to  the  knife-edge.  The  pend- 
ulum was  mounted  on  a  massive  pier  in  the  basement  of  the 
Astrophysical  Observatory  in  Potsdam,  and  was  protected  from 
air  currents  by  a  cloth-lined  wooden  covering. 

The  attracting  masses  were  cast-iron  cylinders  each  weighing 
325  kg.  They  were  so  arranged  on  a  continuous  string  passing 
over  pulleys  that  when  one  was  opposite  the  lower  brass  ball  on 
one  side  of  the  pendulum  the  other  was  opposite  the  upper  ball 
on  the  other  side.  Their  relative  positions  could  be  quickly 
changed  from  without  the  room,  so  that  the  former  mass  came 
opposite  the  upper  ball  and  the  latter  mass  opposite  the  lower  ; 
the  deflection  was  now  in  the  opposite  direction  from  what  it 
was  in  the  first  case. 

131 


MEMOIRS    ON 

The  double  deflection  due  to  the  change  in  position  of  the 
masses,  and  the  time  of  vibration  are  the  quantities  required 
for  the  determination  of  A.  The  readings  for  these  quantities 
were  made  by  the  method  of  Baily,  winch  has  been  already  de- 
scribed. The  time  of  vibration  was  determined  first  with  the 
discs  on  top  of  the  upper  ball,  then  with  one  removed  and  then 
with  still  another  removed.  In  this  way  the  moment  of  inertia 
was  obtained.  The  theory  of  the  instrument  is  complicated, 
and  for  it  reference  must  be  made  to  the  original  paper.  The 
result  obtained  for  A  was  5. 594 ±.032. 

In  1889,  Wilsing  published  (172)  an  account  of  some  further 
observations  with  the  same  apparatus,  some  slight  changes 
having  been  made  in  it  in  the  meantime.  Extra  precautions 
were  taken  in  order  to  avoid  the  effects  of  variations  of  tem- 
perature. Experiments  were  made  with  the  old  balls,  with  new 
lead  balls,  and  with  the  pendulum  rod  alone.  The  mean  re- 
sult from  these  was  5. 588 ±.013  ;  and  the  final  average  of  all 
his  determinations  5.579±.012. 

A  preliminary  paper  (163)  was  read  by  Wilsing  before  the 
Berlin  Academy,  and  also  an  extract  (169)  of  his  first  paper. 
A  condensed  translation  of  both  papers  was  made  by  Prof.  J. 
H.  Gore  (171)  for  the  Smithsonian  Report  for  1888,  and  a  short 
account  of  the  work  is  given  by  Poynting  (185,  pp.  65-9)  and 
by  Fresdorf  (186£,  p.  28). 

POYNTING.  Prof.  J.  H.  Poynting  published  in  1878  the 
results  (146)  of  a  study  of  the  beam  balance.  He  found  that 
the  sources  of  error  were  temperature  changes  producing  con- 
vection currents  and  unequal  expansion  of  the  arms,  and  the 
necessity  of  frequently  raising  the  knife-edges  from  the  planes. 
He  tried  to  overcome  the  former  difficulty  by  taking  the  same 
precautions  as  those  employed  by  users  of  the  torsion  balance  ; 
and  he  did  away  altogether  with  the  raising  of  the  beam  be- 
tween weighings,  and  when  the  weights  had  to  be  exchanged 
held  the  pan  fixed  in  a  clamp. 

The  paper  gives  a  description  of  his  balance  and  illustrates 
how  it  can  be  used,  (I)  to  compare  two  weights,  and  (2)  to  find 
the  mean  density  of  the  earth.  The  motion  of  the  beam  was 
observed  by  means  of  a  telescope  and  scale,  the  mirror  being 
fixed  at  the  centre  of  the  beam.  The  deflection  of  the  ray 
could  be  multiplied  by  repeated  reflections  between  tiiis  mirror 

132 


THE    LAWS    OF    GRAVITATION 

and  another  which  was  fixed  and  nearly  parallel  to  the  former. 
The  centre  of  oscillation  was  determined  after  the  method  of 
Baily  with  the  torsion  balance.  As  a  result  of  11  observations 
Prof.  Poynting  found  the  mean  density  of  the  earth  to  be 
5.69±.15.  He  felt  justified,  therefore,  in  proceeding  to  have  a 
more  suitable  balance  constructed  in  order  to  make  a  more 
careful  determination  of  this  quantity. 

The  investigation  continued  through  many  years,  and  the 
results  (180  and  185,  pp.  71-156)  were  not  published  until  1891. 
Many  unforeseen  difficulties  arose  during  the  progress  of  the 
work,  but  by  patience  and  skill  Poynting  was  able  to  overcome 
these  difficulties  and  to  begin  to  take  observations  in  1890. 
The  balance  was  of  the  large  bullion  type,  123  cm.  long,  and 
made  with  extra  rigidity  by  Oertling.  It  was  set  up  in  a  base- 
ment room  at  Mason  College,  Birmingham.  The  principle 
upon  which  the  experiment  is  based  is  as  follows  :  two  balls 
of  about  the  same  mass  are  suspended  from  the  two  arms  of 
the  balance.  Beneath  the  balance  is  a  turn-table  carrying  a 
heavy  spherical  mass  vertically  under  one  of  the  balls.  The 
position  of  the  beam  is  observed  and  the  turn-table  moved  un- 
til the  mass  is  under  the  other  ball  and  the  position  of  the  beam 
again  observed.  The  deflection  measures  twice  the  attraction 
of  the  mass  for  the  ball.  The  attraction  of  the  mass  for  the 
beam  and  wires,  etc.,  is  then  eliminated  by  repeating  these  ob- 
servations with  the  balls  suspended  at  a  different  distance  be- 
low the  arm,  for  then  the  attraction  of  the  mass  on  the  balance 
remains  the  same,  and  we  find  the  change  in  the  attraction  of 
the  mass  for  the  ball  with  change  of  distance.  The  calculation 
was  complicated  by  the  presence  on  the  tnrn-table  of  another 
mass  as  a  counterpoise  to  the  former  one  ;  it  was  smaller  than 
that  one  and  at  a  correspondingly  greater  distance  from  the 
centre.  It  was  used  because  certain  anomalies  could  ,be  ac- 
counted for  only  on  the  supposition  that  the  floor  tilted  when 
the  turn-table  was  rotated  with  the  large  mass  only  upon  it. 

Instead  of  the  ordinary  mirror  fixed  on  the  beam,  Poynting 
used  the  double-suspension  mirror  (see  Darwin  B.  A.  Rep., 
1881).  The  riders  were  manipulated  by  mechanism  from  with- 
out,  and  the  observer  was  stationed  in  the  room  above,  whence 
he  could  make  all  changes  and  observations  without  opening 
the  balance  room.  The  attracted  and  attracting  masses  were 
made  of  an  alloy  of  lead  and  antimony.  The  balls  were  gilded 

133 


MEMOIRS    ON 

and  weighed  over  21  000  gr.  each.  The  large  mass  weighed 
150  000  gr.,  and  the  counterpoise  about  half  as  much.  A  first 
set  of  observations  gave  A  =  5. 52.  The  attracting  bodies  were 
then  all  inverted  in  order  to  eliminate  the  effects  of  want  of 
symmetry  in  the  position  of  the  turn-table,  and  of  homoge- 
neity in  the  masses.  A  new  set  of  observations  gave  A  =  5.46. 
The  difference  between  the  results  of  the  two  sets  must  have 
been  caused  by  a  cavity  or  irregular  distribution  of  density  in 
the  large  mass,  and  by  other  experiments  Prof.  Poynting  found 
that  its  centre  of  gravity  was  not  at  its  centre  of  figure,  but 
was  nearly  at  the  place  at  which  his  gravitational  experiments 
would  have  suggested  it  to  be.  The  mean  result  for  A  is  taken 
to  be  5.4934,  and  for  the  gravitation  constant,  G,  6.6984xlO~8. 

Poynting  remarks  that  the  effects  of  convection  currents  are 
greater  in  the  beam  balance  than  in  the  torsion  balance,  since 
the  motion  of  the  former  is  in  a  vertical  plane.  He  thinks 
that  a  balance  of  greatly  reduced  dimensions  would  have  been 
preferable.  The  admirable  way  in  which  Prof.  Poynting  has 
utilized  the  common  balance  for  absolute  measurements  of 
force  caused  the  University  of  Cambridge  to  award  him  the 
Adams  Prize  in  1893. 

For  a  short  account  of  this  work  see  Wallentin  (154J). 

BERGET.  In  1756,  Bouguer  read  before  the  Academy  of 
Sciences  the  results  (9)  of  some  experiments  made  by  him  to 
determine  whether  the  plumb-line  was  affected  by  the  tidal 
motion  of  the  ocean.  He  was  not  able  to  detect  any  such 
effect.  Towards  the  middle  of  last  century  Boscovitch  pro- 
posed (140,  vol.  1,  pp.  314  and  327)  to  place  a  long  pendulum 
in  a  very  high  tower  by  the  edge  of  the  sea,  where  the  height 
of  the  tide  is  very  great,  and  to  observe  the  deviation  due  to 
the  rise  of  the  water,  and  thence  to  calculate  the  mean  density 
of  the  earth.  Von  Zach  suggested  (49,  vol.  1,  p.  17)  a  modification 
of  the  experiment.  Boscovitch  also  proposed  the  use  of  a  reser- 
voir after  the  manner  about  to  be  described,  used  by  Berget. 
In  1804,  Robison,  in  his  "Mechanical  Philosophy"'  vol.  1,  page 
339,  points  out  that  a  very  sensible  effect  on  the  value* of  grav- 
ity might  be  observed  at  Annapolis,  Nova  Scotia,  due  to  the 
very  high  tides  there.  The  theory  of  this  local  influence  is 
given  in  Thomson  and  Tait's  "Natural  Philosophy  "  pt.  II., 
page  389.  Struve  (129)  proposed  to  find  A  from  observations 

134 


THE    LAWS    OF    GRAVITATION 

on  plumb-lines  placed  on  each  side  of  the  Bristol  Channel,.and 
Keller  (168)  calculated  the  deflection  of  the  plumb-line  due 
to  the  draining  of  Lake  Fucino.  In  1893,  M.  Berget  utilized 
this  principle  in  order  to  find  (181)  the  density  of  the  earth. 

He  had  the  use  of  a  lake  of  32  hectares  in  area  in  the  Com- 
mune of  Habay-la-neuve  in  Belgian  Luxembourg.  The  level  of 
the  lake  could  be  lowered  1  m.  in  a  few  hours,  and  as  quickly 
regained.  He  could  thus  introduce  under  his  instrument  a 
practically  infinite  plane  of  matter  whose  attraction  could  be 
calculated  and  observed.  The  apparatus  used  to  measure  the 
attraction  was  the  hydrogen  gravimeter  such  as  Boussingault 
and  Mascart  (Comp.  Rend.,  vol.  95,  pp.  126-8)  used  to  find  the 
diurnal  variation  of  gravity.  The  variation  of  the  column  of 
mercury  was  observed  by  the  interference  fringes  in  vacuo  be- 
tween the  surface  of  the  mercury  and  the  bottom  of  the  tube, 
which  was  worked  optically  plane.  A  first  series  of  observa- 
tions was  made  when  the  lake  was  lowered  50  cm.,  and  another 
when  it  was  lowered  1  m.  A  change  of  1  m.  caused  a  displace- 
ment of  the  mercury  column  of  1.26xlO~6  cm.  The  value  of 
the  gravitation  constant  found  was  6.80xlO~~8,  of  A,  5.41  and 
of  the  mass  of  the  earth  5.85  x  1027  gr. 

M.  Gouy  remarks  (182)  that  such  a  result  would  imply  that 
the  temperature  remained  constant  during  hours  to  -g-o0^  000 
of  a  degree,  which  is  impossible.  Pavilion,  with  the  greatest 
care,  was  able  to  reach  T-<ro"oo-  °f  a  degree  only.  So  that  the 
result  given  by  Berget  can  not  be  so  accurate  as  he  supposed. 

For  n,  short  account  of  the  experiment  see  Fresdorf  (186£, 
pp.  29-30). 

BOYS.  Prof.  C.  V.  Boys  read  before  the  Royal  Society,  in  1889, 
an  important  paper  (175  and  176)  on  the  best  proportions  and 
design  for  the  torsion  balance  as  an  instrument  for  finding  the 
gravitation  constant.  He  shewed  that  the  sensibility  of  the 
apparatus,  if  the  period  of  oscillation  is  always  the  same,  is  in- 
dependent of  the  linear  dimensions  of  the  apparatus  ;  and  re- 
marked that  the  statements  of  Cornu  on  this  point  (page  125) 
are  not  correct.  There  are  great  advantages  to  be  gained  by 
reducing  the  dimensions  of  the  apparatus  of  Cavendish  50  or 
100  times ;  the  main  one  is  that  the  possibility  of  variation  of 
temperature  in  the  apparatus  is  enormously  minimized.  Then, 
too,  the  case  can  be  made  cylindrical  and  corrections  for  its  at- 

135 


MEMOIRS    ON 

traction  avoided.  Until  quartz  fibres  existed  it  would  have 
been  impossible  to  have  made  this  reduction  in  the  dimensions 
of  the 'apparatus  and  retained  the  period  of  5  to  10  minutes. 
The  introduction  of  this  invaluable  new  means  of  suspension 
is  also  due  to  Professor  Boys.  Another  improvement  in  the 
form  of  the  apparatus  devised  by  him  is  the  suspending  of  the 
small  balls  at  different  distances  below  the  arm  (the  masses 
must  be  at  corresponding  levels),  so  that  each  mass  acts  prac- 
tically on  one  ball  only. 

Boys  showed  to  the  Society  a  balance  of  this  design ;  it  had 
an  arm  of  only  13  mm.  in  length,  was  18.7  times  as  sensitive  as 
that  of  Cavendish  and  behaved  very  satisfactorily.  He  pro- 
posed to  prepare  a  balance  of  this  kind  especially  suitable  for 
absolute  determinations  and  capable  of  determining  the  gravi- 
tation constant  to  1  part  in  10  000. 

An  account  of  his  completed  work  (187  and  189)  was  read  in 
1894.  For  the  details  of  this  beautiful  experiment  and  the  in- 
genious way  in  which  the  apparatus  was  designed  the  original 
paper  must  be  consulted.  The  general  design  was  that  of  his 
earlier  apparatus,  but  very  great  attention  was  given  to  the 
minutest  details,  and  especially  to  the  arrangements  for  meas- 
uring the  dimensions.  Some  idea  of  the  accuracy  aimed  at 
may  be  got  from  considering  that  in  order  to  obtain  a  result 
correct  to  1  in  10  000  it  was  necessary  to  measure  the  large 
masses  to  1  in  10  000,  the  times  to  1  in  20  000,  some  lengths  to 
1  in  20  000  and  angles  to  1  in  10  000.  The  dimensions  finally 
used  were,  diameter  of  masses  2.25  and  4.25  in.;  distance  be- 
tween masses  in  plan  4  and  6  in.  ;  distance  between  balls  in 
plan  1  in. ;  diameter  of  balls  .2  and  .25  in.  ;  difference  of  level 
between  upper  and  lower  balls  6  in.  The  masses  were  of  lead 
formed  under  great  pressure,  and  the  balls  of  gold. 

The  moment  of  inertia  of  the  beam  was  determined  by  find- 
ing the  period  when  the  balls  were  suspended  from  it,  and 
when  they  were  taken  away  and  a  cylindrical  body  of  silver, 
equal  in  weight  to  the  balls  with  their  attachments,  suspended 
from  the  middle  of  the  beam.  The  apparatus  was  enclosed  by 
a  series  of  metallic  screens  to  prevent  temperature  changes, 
and  outside  of  all  was  a  double -walled  wooden  box  with  the 
space  between  the  walls  filled  with  cotton- wool.  The  final  result 
for  the  gravitation  constant  was  6.6576  x  10~8,  and  for  A,  5.5270. 
The  last  figure  in  each  case  has  no  significance,  but  Boys  con- 

136 


THE    LAWS    OF    GRAVITATION 

sidered  that  the  next  to  the  last  could  not  be  more  than  2  in 
error  at  the  outside.  He  is  still  convinced  that  1  part  in 
10  000  can  be  reached,  but  would  increase  the  length  of  the 
beam  to  5  cm.,  since  the  disturbing  moments  due  to  convection 
are  proportional  to  the  5th  power  of  the  linear  dimensions,  not 
to  the  7th  as  he  had  originally  supposed.  An  excellent  resume 
of  the  experiment  is  to  be  found  in  the  lecture  delivered  by 
Boys  before  the  Royal  Institution  (188). 

EOETVOES.  A  series  of  investigations  upon  gravitation  is  now 
under  way  by  Prof.  R.  von  Eotvos  of  Budapest.  He  has  pub- 
lished a  preliminary  account  (192)  only  of  his  experiments,  but 
they  promise  to  be  very  elaborate  and  exhaustive.  His  paper 
begins  with  a  mathematical  discussion  of  the  space -variation 
of  gravity  as  deduced  from  the  potential  function.  He  investi- 
gates the  equipotential  surface  and  the  measurements  necessary 
to  determine  the  principal  radii  of  curvature,  the  variation  of 
gravity  along  the  surface,  and  the  variation  perpendicular  to  the 
surface.  The  latter  has  already  been  measured  with  the  pend- 
ulum, and  by  Jolly  (144  and  145),  Keller  (152),  Thiesen  (179) 
and  others  with  the  common  balance.  For  the  measurement 
of  the  other  quantities  von  Eotvos  uses  the  torsion  balance. 
This  he  makes  in  two  forms :  the  first  is  of  the  same  general 
type  as  that  of  Baily  and  is  called  the  Krummungsvariometer, 
since  it  is  used  to  measure  the  difference  of  the  reciprocals  of 
the  principal  radii  of  curvature ;  the  second  is  like  that  of 
Boys  in  that  one  ball  is  on  one  end  of  the  rod  and  the  other 
suspended  100  cm.  below  the  other  end  by  means  of  a  wire, 
and  is  called  the  Horizontalvariometer.  The  peculiarity  of 
these  instruments  is  in  the  method  devised  for  getting  rid  of 
convection  currents ;  von  Eotvos  makes  the  case  with  double 
walls  of  thin  metal  with  an  air-space  of  from  5  to  10  mm.  So 
steady  is  the  motion  that  the  balance  can  be  used  in  any  room 
in  the  laboratory,  and  even  in  the  free  air  at  night.  The  period 
is  usually  from  10  to  20  minutes,  and  the  suspension  wire  is  of 
platinum  of  100  to  150  cm.  length.  The  rod  swings  in  a  flat 
cylindrical  box  40  cm.  in  diameter  and  2  cm.  deep. 

Some  investigations  have  been  made  of  the  variation  of 
gravity  in  the  neighbourhood  of  the  hill  Saghberg,  where  von 
Sterneck  found  great  peculiarities..  Some  preliminary  deter- 
minations have  been  made  of  the  constant  pf  gravitation  also, 

137 


MEMOIRS    ON 

with  a  result  6.65xlO~8.  Von  Eotvos  speaks  of  the  method 
employed  HS  an  entirely  new  one,  but  it  is  only  a  variation  of 
that  already  employed  by  Reich,  and  later  by  Dr.  Braun,  the 
oscillation  method.  The  instrument  (the  Krummungsvariom- 
eter)  is  set  up  between  two  pillars  of  lead,  and  the  time  of  vib- 
ration observed  both  when  the  torsion  rod  is  in  the  line  joining 
the  pillars  and  when  it  is  perpendicular  to  this  line.  The 
paper  is  characterized  by  an  almost  total  disregard  of  the  work 
already  done  in  the  field  of  gravitation. 

Eotvos  gives  a  description  of  two  new  instruments  for  use 
in  the  study  of  gravitation.  One  he  calls  the  Gravitationcom- 
pensator ;  in  design  it  is  similar  to  the  others,  but  the  arm 
swings  in  a  narrow  tube.  The  tube  is  surrounded  at  each  end 
by  the  compensating  masses  having  the  balls  at  their  centres ; 
these  masses  are  of  the  shape  of  a  disc  with  two  almost  com- 
plete quadrants  taken  away,  just  enough  being  left  to  hold  the 
remaining  two  quadrants  together.  By  orienting  these  masses 
any  amount  of  compensating  attraction  required  can  be  pro- 
duced. The  other  instrument  is  called  the  Gravitationmulti- 
plicator;  underneath  the  torsion  balance  is  a  turn-table  with 
the  attracting  mass ;  when  the  ball  has  reached  its  maximum 
elongation  in  the  direction  of  the  mass,  the  latter  is  suddenly 
moved  to  the  opposite  side,  and  so  on.  From  the  difference 
of  two  successive  elongations,  and  a  knowledge  of  the  damp- 
ing, the  amount  of  the  attraction  can  be  determined.  This 
is  rather  similar  to  a  piece  of  apparatus  proposed  by  Joly  (177), 
in  1890. 

BRAtnsr.  One  of  the  latest  and  most  elaborate  determinations 
of  the  mean  density  of  the  earth  is  that  made  by  Dr.  Carl 
Braun,  S.J.,  at  Mariaschein  in  Bohemia  (193).  In  its  gen- 
eral form  the  apparatus  is  like  that  employed  by  Reich  in  his 
later  experiments.  Like  Reich,  too,  he  uses  two  distinct 
methods  of  finding  his  results,  the  deflection  and  the  oscilla- 
tion methods.  The  experiments  of  Dr.  Braun  differ  however 
in  several  very  important  respects  from  those  of  Reich ;  the 
dimensions  of  the  apparatus  are  much  reduced,  the  masses  are 
suspended  from  wires  and  the  deflection  is  determined  differ- 
ently;  but  the  respect  in  which  it  differs  from  all  previous  de- 
terminations is  in  the  fact  that  the  torsion  rod  swings  in  a 
partial  vacuum  of  about  4  mm.  of  mercury.  This  was  sug- 

138 


THPJ    LAWS    OF    GRAVITATION 

gested  by  Faye  (130)  and  by  Boys,  but  no  investigation  of  the 
kind  had  ever  been  made. 

The  experiments  were  begun  in  1887.  In  a  corner  of  a  liv- 
ing-room a  heavy  stone  slab  was  set  into  the  stone  wall ;  on 
this  was  a  glass  plate  from  which  arose  a  brass  tripod  to  carry 
the  suspension  wire,  1  in.  long,  and  the  torsion-rod  ;  and  on 
the  plate  fitted  airtight  a  conical  glass  cover  within  which  a 
vacuum  could  be  made.  The  apparatus  was  so  tight  that  the 
pressure  inside  did  not  change  in  4  years.  Suspended  from  a 
movable  ring  encircling  the  glass  cover  were  the  two  masses, 
about  42  cm.  apart ;  the  masses  used  were  two  sets  of  spheres, 
one  of  brass  weighing  5  kg.  each,  and  the  other  of  iron,  112  mm. 
in  diameter,  filled  with  mercury,  and  weighing  about  9.15  kg. 
each.  The  torsion  rod  was  a  triangle  of  copper  wires  and  the 
balls  were  suspended  from  its  ends  and  lay  in  the  same  hori- 
zontal plane  24.6  cm.  apart.  Each  ball  was  of  gilded  brass  and 
weighed  about  54  gr.  In  order  to  provide  against  temperature 
changes,  the  whole  apparatus  was  surrounded  by  metal  screens, 
cloth  hangings  and  wooden  enclosures. 

The  deflection  method  of  observation  is  practically  that  of 
Cavendish,  but  the  position  of  the  centre  was  found  rather 
differently.  Dr.  Braun  observed  the  time  of  the  passage  in 
each  direction  across  the  wire  of  the  telescope  of  several  scale 
divisions  near  the  centre,  and  took  as  the  centre  that  point 
with  reference  to  which  the  time  of  oscillation  was  the  same  in 
both  directions.  He  found  for  the  final  corrected  mean  result 
A  =  5. 52962. 

In  the  oscillation  method  the  period  was  determined  when 
the  masses  were  in  the  line  joining  the  balls,  and  also  when  they 
were  in  a  line  at  right  angles  to  that  direction.  The  final  cor- 
rected mean  result  gave  A  =  5. 52920.  The  mean  of  all  is 
5.52945 ib. 0017.  The  extremes  were  5.5094  and  5.5511.  The 
mean  of  all  the  results  found  in  1892  was  5.52770,  and  in  1894 
was  5.53048. 

The  final  result  for  the  gravitation  constant  was  6.655213 

xio-8. 

In  an  appendix  is  given  a  further  discussion  of  the  correc- 
tions to  be  made  on  account  of  damping.  Dr.  Braun  found  his 
former  estimate  to  be  in  error,  and  after  examination  gave  as 
the  most  probable  final  results,  A  =  5. 52700 ±.0014,  and  Gr 
=  (6.65816±.  00168)  x  10~8. 

139 


MEMOIRS    ON 
A  concise  account  of  the  work  is  given  in  Nature  (197). 

KOENIG,    RlCHARZ    A^D     KRIGAR  -  MENZEL.       Ill  1884,  Pro- 

fessors  A.  Konig  and  F.  Richarz  proposed  (159  and  160)  to 
determine  the  gravitation  constant  by  a  method  which  is  a 
modification  of  that  used  by  Jolly  (page  126).  In  the  latter 
experiment  the  lower  set  of  scale  pans  was  21  m.  beneath  the 
upper  aud  differences  of  temperature  were  unavoidable.  The 
improvement  proposed  was  to  have  the  sets  of  scale  pans  much 
closer  together,  to  measure  the  change  of  weight  with  height 
after  the  manner  of  Jolly  and  then  to  insert  between  the  upper 
and  lower  pans  a  huge  block  of  lead  with  holes  in  it  for  the 
passage  of  the  wires  supporting  the  lower  pans.  A  weighing 
was  made  with  two  nearly  equal  masses,  one  in  the  right  upper, 
the  other  in  the  left  lower  pan  ;  then  the  former  in  the  right 
lower  is  balanced  against  the  latter  in  the  left  upper  pan.  From 
these  weighings,  taking  account  of  the  result  of  similar  ob- 
servations without  the  block,  the  value  of  4  times  the  attrac- 
tion of  the  block  is  determined,  and  from  a  comparison  of  this 
result  with  the  calculated  attraction,  the  gravitation  constant 
can  be  determined.  Professors  Konig  and  Richarz  seem  to 
have  hit  upon  the  same  idea  independently  of  each  other.  In 
1881,  Keller  proposed  (167)  a  somewhat  similar  modification  of 
the  Jolly  experiment.  Professor  A.  M.  Mayer  suggested  (161) 
the  use  of  mercury  instead  of  lead  for  the  attracting  mass,  but 
Konig  and  Richarz  replied  (162)  that  Mayer  had  misunder- 
stood the  form  of  the  experiment,  and  gave  a  lucid  and  simple 
explanation  of  their  method. 

In  1893,  appeared  a  report  (183  and  184)  on  the  observations 
made  to  find  the  decrease  of  gravity  with  increase  of  height. 
A  description  is  given  of  the  balance  and  of  the  improvements 
introduced  into  it  in  order  to  overcome  the  liability  to  varia- 
tion in  its  readings.  The  masses  weighed  against  each  other 
were  1  kg.  each,  and  the  balance  had  a  sensitiveness  of  1  part 
iirl  000  000.  All  exchange  of  weights  was  made  automatically 
without  opening  the  covers.  The  apparatus  was  carefully  sur- 
rounded with  metal  screens  to  ward  off  temperature  changes. 
It  was  set  up  in  a  bastion  of  the  citadel  at  Spandau,  and  in 
consequence  of  the  departure  of  Konig  to  accept  a  professor- 
ship at  Berlin,  Dr.  Krigar-Menzel  assisted  in  the  carrying  on 
of  the  research.  The  pans  were  2.26  m.  apart  vertically.  The 

140 


THE    LAWS    OF    GRAVITATION 

change  in  gravity  observed  was  .000006523   -—  -  ;  whereas  the 

sec.2' 

calculated  value  was  .00000697.  The  difference  is  ascribed  to 
the  local  attraction  of  walls,  etc. 

The  paper  '198)  embodying  the  final  results  was  presented 
to  the  Berlin  Academy  in  Dec.,  1897.  A  most  exhaustive  ex- 
amination had  been  made  of  the  possible  sources  of  error,  and 
the  devices  for  overcoming  these  difficulties  were  most  ingen- 
ious and  elaborate.  In  the  cases  where  the  sources  of  error 
could  not  be  eliminated,  as  in  the  variations  of  temperature  with 
time  and  place,  the  effect  is  carefully  considered  and  allowed 
for.  Observations  were  made  continuously  from  Sept.,  1890,  to 
Feb.,  1896,  and  from  the  elegance  of  the  method  and  the  time 
and  care  devoted  to  the  working  out  of  the  result,  this  deter- 
mination of  the  gravitation  constant  and  the  mean  density  of 
the  earth  must  be  taken  as  one  of  the  very  best. 

The  block  of  lead  weighed  100  000  kg.,  was  200  cm.  high  and 
210  cm.  square  and  was  built  up  out  of  bars  of  lead  10  x  10  x30 
cm.  on  the  top  of  a  massive  pier.  The  amount  of  the  settling 
of  the  pier  was  measured  and  found  to  be  not  important,  and 
the  shape  of  the  block  was  not  distorted  by  its  own  pressure. 
The  final  value  for  G  was  (6.685  ±.011)  10~8,  and  for  A,  5.505 
±.009. 

Professors  Richarz  and  Krigar-Menzel  published  (191  and 
199),  in  1896,  a  condensed  account  of  their  work.  Other  ab- 
stracts are  also  to  be  found  (185,  pp.  64-5,  186J,  pp.  26-7,  and 
194). 


NOTICES'  In  1889,  Dr.  W.  Laska  of  Prague  proposed 
(173)  a  method  of  finding  the  density  of  the  earth.  At  the  top 
of  a  rod  projecting  above  a  pendulum  is  a  lens  which  is  so  close 
to  a  fixed  plate  of  glass  that  Newton's  rings  are  visible.  A 
hollow  ball  near  the  bob  of  the  pendulum  is  then  filled  with 
mercury  and  attracts  the  bob,  bringing  the  lens  nearer  the 
plate  ;  an  observation  of  the  movement  of  the  Newton's  rings 
will  measure  the  deflection  of  the  bob.  No  further  report  has 
been  published.  An  account  of  his  method  is  given  by  Gflnther 
(196J,  vol.  1,  p.  197). 

About  the  same  time  Professor  Joly  of  Dublin  suggested 
(177)  a  resonance  method  for  the  same  purpose.  A  pendulum  in 
a  vacuous  vessel  has  the  same  period  as  two  massive  ones  kept 

141 


MEMOIRS    ON 

going  outside  the  vessel.  The  amplitude  of  the  motion  of  the 
inner  pendulum  due  to  a  given  number  of  swings  of  the  outer 
ones  would  give  a  measure  of  the  constant  of  gravitation. 

In  1895,  Professor  A.  S.  Mackenzie  of  Bryn  Mawr  College 
published  an  account  (190)  of  some  experiments  with  the  Boys' 
form  of  torsion  balance  to  determine  whether  the  gravitational 
properties  of  crystals  vary  with  direction.  No  such  variation 
was  found  in  the  case  of  calc-spar,  the  crystal  under  investiga- 
tion. He  shewed  further  that  the  inverse  square  law  holds 
good  in  the  neighbourhood  of  a  crystal  to  one-fifth  per  cent. 

Two  years  later  appeared  an  account  (196)  of  an  investigation 
by  Professors  Austin  and  Thwing  of  the  University  of  Wiscon- 
sin to  determine  whether  gravitational  attraction  is  indepen- 
dent of  the  intervening  medium,  that  is,  whether  there  is  a 
gravitational  permeability.  No  effect  was  found  due  to  the 
medium  within  the  limits  of  error  of  the  method. 

At  a  meeting  of  the  "  Deutcher  Naturforscher  und  Aerzte" 
in  Brunswick,  in  1897,  Professor  Drtide  read  a  paper  (195)  on 
action  at  a  distance,  which  contains  a  very  valuable  account  of 
the  theory  of  gravitation,  and  should  be  consulted  by  any  one 
wishing  to  find  a  brief  resume  of  that  subject,  and  especially 
for  a  discussion  of  the  velocity  of  propagation  of  gravitation. 

The  latest  work  on  the  laws  of  gravitation  is  that  of  Profess- 
ors Poynting  and  Gray  (200)  on  the  search  for  a  directive 
action  of  one  quartz  crystal  on  another.  A  small  crystal  was 
suspended  and  its  time  of  rotative  vibration  noted  ;  a  large 
crystal  in  the  same  horizontal  plane  was  then  rotated  about 
a  vertical  axis  through  its  centre  with  a  period  either  equal  to, 
or  twice,  that  of  the  smaller  crystal.  If  there  were  any  directive 
action  the  small  crystal  should  be  set  in  vibration  by  forced 
oscillations;  no  such  effect  was  found. 


142 


THE    LAWS    OF    GRAVITATOIN 


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verend Pere  Bertier.  [Rozier]  Journ.  de  Phys., 
2,  378-81. 

23  1774    J.  P.  David  and  Fathers  Cotte  and  Bertier.    (Notice  of  their  ex- 

periments).    [Rozier]  Journ.  de  Phys.,  4,  338. 

24  1774    J.  P.  David.       Reponse  aux  reflexions  de  M.  Lesage.     [Rozier} 

Journ.  de  Phys.,  4,  431-41. 

25  1774    Abbe  Rozier.      Observations   sur   la   lettre    de    Pere    Bertier. 

[Rozier]  Journ.  de  Phys.,  4,  454-61. 

26*  1774    Father  Bertier.  (Account  of  experiments).     Journ.  de  Verdun, 
148-185. 

27  1775    J.  P.  David.       Sur  la  pesanteur  des  corps.     [Rozier}  Journ.  de 

Phys.,  5,  129-139. 

28  1775    Father  Bertier.  (Letter).     [Rozier]  Journ.  de  Phys.,  5,  305-13. 

29  1775    —  (Account  of  exp'ts.  made  by  com.  of  Acad.  of 

Dijon).     [Rozier]  Journ.  de  Phys.,  5,  314-26. 

30  1775    Chev.  de  Dolomieu.    Experiences  sur  la  pesanteur  des  corps  a 

differentes  distances  du  centre  de  la  terre. 
[Rozier]  Journ.  de  Phys.,  6,  1-5. 

31  1775    N.  Maskelyne.  A  proposal  for  measuring  the  attraction  of  some 

hill  in  this  kingdom  by  astronomical  observa- 
tions. Phil.  Trans.  Lond.,  495-9. 

32  1775    N.  Maskelyne.  An  account  of  observations  made  on  the  moun- 

tain Schehallien  for  rinding  its  attraction.  Phil. 
Trans.  Lond.,  500-42. 

33  1776    F.  K.  Achard.    Bemerkungen  ilber  die  von  Herrn  Bertier  an- 

gestellten  Versuche.  Beschaft.  der  Berl.  Oes. 
Naturf.  Freunde,  2,  1-11. 

34  1776    G.  L.  Lesage.     Experiences  et  vues  sur  1'intensite  de  la  pes- 

anteur dans  1'interieur  de  la  terre.  [Rozier] 
Journ.  de  Phys.,  7,  1-12. 

35  1776    J.  Pringle.          Discours  sur  I'attraction  des  montagnes ;  traduit 

par  M.  le  Roy.  [Rozier]  Journ.  de  Phys.,  7, 
418-34. 

36  1777    Father  Bertier.  Retraction  du  Pere  Bertier  de  1'Oratoire,  sur  la 

consequence  qti'il  a  tire  de  son  experience  d'un 
corps,  pesant  plus  dans  un  lieu  haut  que  dans 
un  bas.  [Rozier]  Journ.  de  Phys.,  9,  460-6. 

37  1779    C.  Button.          An  account  of  the  calculations  made  from  the 

survey  and  measures  taken  at  Schehallien  in 
order  to  ascertain  the  mean  density  of  the  earth. 
Phil.  Trans.  Lond.,  68,  689-788. 

38  1780    C.  Hutton.          Calculations  to  determine  at  what  point  in  the 

side  of  a  hill  its  attraction  will  be  the  greatest. 
Phil.  Trans.  Lond.,  1-14. 

39  1798    H.  Cavendish.    Experiments  to  determine  the  density  of  the 

earth.     Phil.  Trans.  Lond.,  88,  469-526. 
146 


THE    LAWS    OF    GRAVITATION 

394  1799    —  (Review  of  39).    Bibl.  Brit.,  11,  233-41. 

40  1799    L.  W.  Gilbert.       Versuche,  urn  die  Dichtigkeit  zu  bestimmen, 

von  Henry  Cavendish,  Esq.  [Gilbert]  Ann.  der 
Phys.,  2,  1-62. 

41  1803    A.  Motte.  The   mathematical  principles  of  natural   phi- 

losophy by  Sir  I.  Newton,  translated  into  English 
by  Andrew  Motte,  to  which  are  added  Newton's 
system  of  the  world,  etc.  W.  Davis'  edn.  3 
vol.  London.  8V0. 

42  1806     H.  W.  Brandes.     Theoretische  Untersuchungen  liber  die  Oscilla- 

tionen  der  Drehwaage  bei  Cavendish's  Ver- 
suchen  ilber  die  Attraction  kleiner  Massen. 
Mag.  fur  den  Neuesten  Zustand  der  Naturkunde, 
12,  300-310. 

43  1809    F.  X.  von  Zach.       Ueber  die  Moglichkeit  die  Gestalt  der  Erde 

aus  Gradmessungen  zu  bestimmen.  Monatl. 
Corresp. ,  2O,  3-9. 

44  1810    F.  X.  von  Zach.       Ueber  Densitat  der  Erde  und  deren  Eiufluss 

auf  geographische  Ortsbestimmungen.  Monatl. 
Corresp.  ,21,  293-310. 

45  1811     C.  Hutton.  On   the  calculations  for  ascertaining  the  mean 

density  of  the  earth.  [Tilloch~]  Phil.  Mag.,  38, 
112-6. 

46  1811    J.  Playfair.         Account  of  a  lithological  survey  of  Schehallien, 

made  in  order  to  determine  the  specific  gravity 
of  the  rocks  which  compose  that  mountain. 
Phil.  Trans.  Lond.,  347-77. 

47  1812    C.  Hulton.          Tracts  on  mathematical  and  philosophical  sub- 

jects.    3  vol.     London.     8V0. 

48  1813    L.  W.  Gilbert.  Bericht  von  einer  lithologischen  Aufnahme  des 

Schehallien,  urn  das  specifische  Gewicht  der 
Gebirgsarten  desselben,  und  daraus  die  mittlere 
Dichtigkeit  der  Erde  zu  bestimmen,  von  J.  Play- 
fair,  Esq.  Pogg.  Ann.,  43,  62-75. 

49  1814    F.  X.  von  Zach.  L'attraction  des  montagnes,  et  ses  effets  sur  les 

fils  si  plomb  ou  sur  les  niveaux  des  instruments 
d'astronomie.  2  vol.  Avignon.  4to. 

50  1815    N.  M.  Chompre.    Experiences  pour  determiner  la  densite  de  la 

terre  ;  par  Henry  Cavendish.  Traduit  de  1'An- 
glais.  Journ.  de  V&c.  Roy.  Poly  technique.  Cahier 
17.  1C,  263-320. 

51  1819    T.  Young.  Remarks  on  the  probabilities  of  error  in  physical 

observations,  and  on  the  density  of  the  earth. 
Phil.  Trans.  Lond.,  70-95  ;  Mine.  Works,  2,  8-28. 

52  1820    C.  Hutton.          (Letter  to  Laplace).  [Blainville]  Journ.  dePhys., 

9O,  307-12. 

53  1821     C.  Hutton.          On  the  mean  density  of  the  earth.     [Tilloch] 

Phil.  Mag.,  58,  3-13. 

54  1821     C.  Hutton.          (Same  title  as 53).     Phil.  Trans.  Lond.,  276-292. 

147 


MEMOIRS    ON 

55  1824    F.  Carlini.          Osservazioni  della  lunghezza  del  pendolo  sem- 

plicefatte  all'  altezza  di  mille  tese  sul  livellodel 
mare.  Eff.  Astr.  di  Milano,  a  pp.  28-40. 

56  1825    S.  (Notice  of  55).     [ferussac]  Bull,  des  Sc.  IJath., 

3,298-301. 

57  1826    M.  W.  Drobisch.    De  vera  lunae  figura.     Lipsiae.     12mo. 

58  1827    E.  S(abine).         An  account  of  Prof.  Carlini's  experiments  on 

Mont.-Cenis.     Quart.  Journ.  of  Sc.,  24,  153-9. 

59  1827    M.  W.  Drobisch.    Ueber  die  in  den  Minen  von  Dolcoatb  in  Corn- 

wall neuerlich  angestellten  Pendelbeobachtung- 
en.  Pogg.  Ann.,  1O,  444-456. 

60  1827    -  (Notice  of  Dolcoath  expt.).     Phil.  Mag.,  [2],  1, 

385-6. 

61  1825-45    J.  S.  T.  Gebler.     Physikalisches  Worterbuch.    22vol.     Leip- 

zig.    8V0. 

62*  1828    -  Account  of  experiments  made  at  Dolcoath  mine 

in  Cornwall,  in  1826  :md  1828,  for  the  purpose 
of  determining  the  density  of  the  earth.  Cam- 
bridge. 8V0.  Printed  privately. 

63  1828    M.  W.  Drobisch.     Ausfilhrlicher  Bericht  liber  mehrere  in  den 

Jahren  1826  und  1828  in  den  Minen  von  Dolcoath 
in  Cornwall  zur  Bestimmung  der  mittleren  Dich- 
tigkeit  der  Erde  angestellte  Pendelversuche. 
Pogg.  Ann.,  14,  409-27. 

64  1829-30    J.  C.  E.  Schmidt.     Lehrbuch  der  mathematischen  und  phy- 

sischen  Geogr;ipliie.     2  vol.     Gottingen.     8V0. 

65  1833     S.  D.  Poisson.     Traitede  mecanique.    2d  edn.    2vol.    Paris.  8V0. 

66  1837    E.  de  Beaumont.     Extiaitd  un  memoire  de  M.  Reich  sur  la  den- 

site  de  la  terre.     Camp.  Rend.,  5,  697-700. 

67  1838    F.Reich.  Versuche  iiber  die  mittlere  Dichtigkeit  der  Erde 

mittelst  der  Drehwage.     Freiberg.     8V0. 

68  1838    -  On  the  repetition  of  the  Cavendish  experiment, 

for  determining  the  mean  density  of  the  earth. 
Phil.  Mag.,\S\,  12,  283-4. 

69  1839    F  Baily.  (Same  title  as  68).     Mon.  Not.  Roy.  Astr.  Soc., 

4,  96-7. 

70  1840    C.  I.  Giulio.        Sur  la  determination  de  la  densite  moyenne  de 

la  terre,  dednite  de  1'observation  du  peudule  faite 
&  1'Hospice  du  Mont-Cenis  par  M.  Carlini  en 
Septcmbre,  1821.  Mem.  Accad.  Torino,  [2],  2, 
379-84. 

71  1840    L.  F.  Menabrea.    Calm]  de  la  densite  de  la  terre.     Mem.  Accad. 

Torino,  [2],  2,  305-68. 

72  1840     A.  G.  Calcul  de  la  densite  de  la  terre,  par  L.  F.  Mena- 

brea.   Bibl.  Unit),  de  Geneve,  [nouv.],  27,  163-75. 

73  1841     L.  F.  Menabrea.    On  Cavendish's  experiment.     Phil .  W((j.t  [3], 

19,  62-3. 

74  1842    J.  F.  Saigey.      Densite  du  globe.     Rev.  Scient.  et  Ind.,  [Queme- 

mlle],  1 1.  149-60  and  242-53,  and  12,  373-88. 
148 


THE    LAWS    OF    GRAVITATION 


75  1842 

76  1842 


F.  Daily. 
F.  Baily. 


77     1842    F.  Baily. 


78  1843 

79  1843 

80  1843 
80  1845 


F.  Baily. 
F.  Baily. 

A.  G. 

C.  A.  F.  Peters 


81  1847  G.  W.  Hearn. 

82  1849  E.  Sabine. 

83  1852  F.  Reich. 

84  1852  - 

85  1852  Schaar. 

86  1853  — 


87  1853 

88  1855 

89  1855 


F.  Reich. 

G.  B.  Airy. 
G.  B.  Airy. 


90    1855    — 


An  account  of  some  experiments  -with  the  tor- 
sion rod,  for  determining  the  mean  density  of 
the  earth.  Phil.  Mag. ,  [3],  21,  1 11-21. 
Resultats  de  quelques  experiences  failes  avec  la 
balance  de  torsion,  pour  determiner  la  densite 
moyenne  de  la  terre.  Ann.  de  Ghim.  et  de  PJiys., 
[3],  5,  338-53. 

Bericht  von   einigen  Versuchen  mil  der  Dreh- 
wage  zur  Bestirnmung  der  mittleren  Dichtigkeit 
der  Erde.     Pogg.  Ann.,  57,  453-67. 
(Same  title  as  75).     Mon.  Not.  Roy.  Astr.  Soc., 
5,  188  and  197-206. 

Experiments  with  the  torsion-rod  for  determin- 
ing the  mean  density  of  the  earth.      Mem.  Roy. 
Astr.  Soc.,  14,  1-120  and  i.-ccxlviii. 
(Same  title  as 76).  Bibl.  Univ.  de  Geneve,  [nouv.'], 
43,  177-81. 

Von  den  kleinen  Ablenkungen  der  Lothlinie 
und  des  Niveaus,  welche  durch  die  Anziehung- 
en  der  Sonne,  des  Mondes,  und  einiger  terres- 
trischen  Gegenstiinde  hervorgebracht  werden. 
Astr.  Nach.,  22,  33-42. 

On  the  cause  of  the  discrepancies  observed  b}T 
Mr.  Baily  with  the  Cavendish  apparatus  for  de 
termhiirig  the  mean  density  of  the  earth.  Phil. 
Trans.  Lond.,  217-29 

Cosmos,  by  Alexander  von  Humboldt,  trans- 
lated under  the  superintendence  of  Lieut. -Col. 
Edward  Sabine.     6th  Edn.     1.  London.     8V0. 
Neue  Versuche  mil  der  Drehwaage.    Le-ip.  Abh. 
math.  phy.  cl,  1,  383-430. 

Neue  Versuche  liber  die  mittlere  Dichtigkeit  der 
Erde,  von  F.  Reich.  Pogg.  Ann.,  85,  189-98. 
Rapport  de  M.  Schaar  snr  iin  rnemoire  de  M. 
Montigny  relatif  aux  experiences  pour  deter- 
miner la  densite  de  la  terre.  Bull.  Acad.  Roy. 
Belg.,  19,  pt.  2,  476-81. 

Nouvelles  experiences  sur  la  densite  moyenne 
de  la  terre.  Ann.  de  Chim.  et  de  Phys.,  [3],  38, 
382-3. 

New  experiments  on  the  mean  density  of  the 
earth.     Phil.  Mag.,  [4],  5,  154-9. 
(Report  on    Harton  expts.).      Mon.  Not.  Roy., 
Astr.  Soc.,  15,35-6. 

Note  respecting  the  recent  experiments  in  the 
Harton  Colliery.  Mon.  Not.  Roy.  Astr.  Soc., 
15,  46. 

(Report  on    Harton  expts.).      Mon.  Not.   Roy. 
Astr.  Soc.,  15,  125-6. 
149 


MEMOIRS    ON 


91    1855    — 


92    1855    — 


93     1855    J.  H.  Pratt. 


94    1855    G.  B.  Airy. 


95  1855 

96  1856 

97  1856 

98  1856 

99  1856 


T.  Young. 
J.  H.  Pratt. 
G.  B.  Airy. 

J.  H   Pratt. 


100    1856    G.  B.  Airy. 


101     1856    G.  B.  Airy. 


102    1856    G.  G.  Stokes. 


103    1856    H.  James  and 


Note  sur  les  observations  du  pendule  executees 
dans  les  mines  de  Harton  pour  determiner  la 
densite  moyennedela  terre  ;  par  M.  Airy.  Ann. 
de  Chim.  et  de  Phys.,  [3],  43,  381-3. 
Extrait  du  rapport  presente  a  la  35me  seance 
anuiversaire  de  laSociete  Royale  Astronomique 
de  Londres  par  le  conseil  de  cette  societe  le  9 
Fevrier,  1855.  Arch,  des  Sc.  Phys.  et  Nat.,  29, 
188-191. 

On  the  attraction  of  the  Himalaya  mountains, 
and  of  the  elevated  regions  beyond  them,  upon 
the  plumb-line  in  India.  Phil.  Trans.  Lond., 
145,53-100. 

On  the  computation  of  the  effect  of  the  attrac- 
tion of  mountain-masses,  as  disturbing  the  ap- 
parent astronomical  latitude  of  stations  in 
geodetic  surveys.  Phil.  Trans.  Lond.,  145, 
101-4. 

Miscellaneous  works  and  life,  by  Peacock  and 
Leitch.     4  vol.     London.     8V0. 
(Same  title  as  93).     Mon.  Not.  Roy.  Astr.  Soc., 
16,  36-41  and  104-5. 

(Same  title  as  94).  Mon.  Not.  Roy.  Astr.  Soc., 
16,  42-43. 

(Report  on  Harton  expts.).  Mon.  Not.  Roy. 
Astr.  Soc.,  16,  104. 

On  the  effect  of  local  attraction  upon  the  plumb- 
line  at  stations  on  the  English  arc  of  the  merid- 
ian, between  Dunnose  and  Burleigh  Moor  ;  and 
a  method  of  computing  its  amount.  Phil.  Trans. 
Lond.,  146,  31-52, 

A.ccount  of  pendulum  experiments  undertaken 
in  the  Harton  Colliery,  for  the  purpose  of  de- 
termining the  mean  density  of  the  earth.  Phil. 
Trans.  Lond.,  146,  297-342. 
Supplement  to  the  "account  of  pendulum  ex- 
periments undertaken  in  the  Harton  Colliery"  ; 
being  an  account  of  experiments  undertaken 
to  determine  the  correction  for  the  temperature 
of  the  pendulum.  Phil.  Trans.  Lond.,  146, 
343-55. 

(Addendum  to  101  ;  on  the  effect  of  the  earth's 
rotation  and  ellipticity  in  modifying  the  numer- 
ical results  of  the  Harton  experiment).  Phil. 
Trans.  Lond.,  146,  353-5. 
A.  R.  Clarke.  On  the  deflection  of  the  plumb- 
line  at  Arthur's  Seat,  and  the  mean  specific 
gravity  of  the  earth.  Phil.  Trans.  Lond.,  146, 
591-606. 

150 


THE    LAWS    OF    GRAVITATION 


104  1856  H,  James. 


105  1856  — 


106  1856  S.  Haughton. 


107  1856  G.  B.  Airy. 

108  1856  H.  James. 


109  1856  G.  B.  Airy. 

110  1856  — 


111  1856  G.  B.  Airy. 


112  1857  E.  R. 


113  1857  — 


114  1857  — 

115  1857  — 

116  1857  — 

117  1857  H.  James. 

118  1857  W.  S.  Jacob. 

119  1857  G.  B.  Airy. 

120  1857  H.  James. 


On  the  figure,  dimensions  and  mean  specific 
gravity  of  the  earth,  as  derived  from  the  ord- 
nance trigonometrical  survey  of  Great  Britain 
and  Ireland.  Phil.  Trans.  Lond.,  146,  607-26. 
Ueber  die  in  der  Kohlengrube  von  Harton  zur 
Bestimmung  der  mittleren  Dichte  der  Erde  un- 
ternominenen  Pendelbeobachtungen  ;  von  G.  B. 
Airy.  Fogg.  Ann.,  97,  599-605. 
On  the  density  of  the  earth,  deduced  from  the 
experiments  of  the  Astronomer  Royal,  in  the 
Harton  coal-pit.  Phil.  Mag.,  [4],  12,  50-1. 
(Same  title  as  100).  Phil.  Mag.,  [4],  12,  226-31. 
Account  of  the  observations  and  computations 
made  for  the  purpose  of  ascertaining  the  amount 
of  the  deflection  of  the  plumb-line  at  Arthur's 
Seat,  and  the  mean  specific  gravity  of  the  earth. 
Phil.  Mag.,  [4],  12,  314-6. 
(Same  title  as  101).  Phil.  Mag.,  [4],  12,  467-8. 
Ueber  die  Dichtigkeit  der  Erde,  hergeleitet  aus 
den  Versuchen  des  Konigl.  Astronomen  (Hrn. 
Airy)  in  der  Kohlengrube  Harton  ;  vonSr.  Ehr- 
wurd.  Samuel  Haughton,  Fellow  des  Trinity 
College  in  Dublin.  Pogg.  Ann.,  99,  332^. 
On  the  pendulum  experiments  lately  made  in 
the  Harton  Colliery,  for  ascertaining  the  mean 
density  of  the  earth.  Am.  Journ.  Sc.,  [2],  21, 
359-64. 

Memoire  sur  les  experiences  enterprises  dans  la 
mine   de    Harton    pour   determiner   la   densite 
moyenne  de  la  terre,  par  G.  B.  Airy.     Arch,  des 
Sc.  Phys.  et  Nat.,  35,  15-29. 
Ueber  die  Dichtigkeit  der  Erde,  hergeleitet  aus 
den  Pendelbeobachtungen   des  Herrn  Airy  in 
der  Kohlengrube  Harton  von  Herrn  S.  Haugh- 
ton, Fellow  am  Trinity-College  in  Dublin.  Zeit. 
fur  Math.  u.  Phys.,  2,  68-70. 
Ueber  die  Bestimmung  der  mittleren  Dichtigkeit 
der  Erde.     Zeit.  fur  Math.  u.  Phys..  2,  128-30. 
(Same  title  as  103).     Proc.  Roy.  Soc.  Edin.,  3, 
364-6. 

(Notice  of  106).     Am.  Journ.  Sc.,  [2],  24,  158. 
(Same  title  as  104).     Phil.  Mag. ,  [4],  1 3, 129-32. 
On  the  causes  of  the  great  variation  among  the 
different  measures  of  the  earth's  mean  density. 
Phil.  Mag.,  [4],  13,  525-8. 
(Same  title  as  101).     Proc.  Roy.  Soc.  Lond.,  8, 
58-9. 

(Same  title  as  104).     Proc.  Roy.  Soc.  Lond.,  8, 
111-6. 

151 


MEMOIRS    ON 

121  1857     W.  S.  Jacob.     (Same  title  as  118).     Proc.  Roy.  Soc.  Lend.,  8, 

295-9. 

122  1858    G.  B.  Airy.        (Same  title  as  111).     Proc.  Roy.  Inst.,  2,  17-22. 

123  1858    —  (Same  title  as  103).     Mon.  Not.  Roy.  Astr.  Soc., 

18,  220. 

124  1858    -  (Same  title  as  104).     Mon.  Not.  Roy.  Astr.  Soc., 

18,  220-2. 

125  1858    H.  James  and  A.  R.  Clarke.    Ordnance  trigonometrical  Survey 

of  Great  Britain  and  Ireland.  Account  of  the 
observations  and  calculations  of  the  principal 
triangul.-ition  ;  and  of  the  figure,  dimensions  and 
mean  specific  gravity  of  the  earth  as  derived 
therefrom.  2  vol.  London.  4to. 

126  1859    —  (Same  title  as  125).     Mon.  Not.  Roy.  Astr.  Soc., 

19,  194-9. 

127  1859    P.  F.  J.  Gosselin.  Nouvelexamen  sur  la  densite  rnoyenne  de  la 

terre.     Mem.  Acacl.  Imp.  de  Mete,  [2],  7,  469-85. 

128*  1859-60  E.  Sergent.  Sulla  densita  della  materia  nell'  intoruo  del 
globo,  e  sulla  potenza  della  crosta  terrestre.  Atti 
della. Soc.  Ital.  di  8c.  Nat.  Milano,  2,  169-175. 

129  1861     O.  Struve.         Ueber  einen  von  General   Schubert  an  die  Aka- 

demie  gerichteten  Antrag  betreffend  die  Rus- 
sisch  -  Scandinavische  Meridian  -  Gradmessung. 
Bull.  Acad.  St.  Petersb.  phys.  math,  d.,  3,  395- 
424. 

130  1863     H.  A.  E.  A.  Faye.  Sur  les  instruments   geodesiques  et  sur  la 

densite  moyenne  de  la  terre.  Comp.  Rend.,  £>G, 
557-66. 

131  1864    E.  Pechmann.  Die    Abweichung    der  Lothlinie    bei  astrono- 

mischen  Beobachtungsstationen  und  ihre  Be- 
rechnung  als  Erforderniss  einer  Gradmessung. 
Denkschr.  Acad.  Wiss.  Wien.  math.-naturw.  cl., 
22,  41-88. 

132  1864    J.  Babinet.         Note  surle  calcul  de  1'experiencede  Cavendish, 

relative  a  la  masse  et  a  la  densite  moyenne  de  la 
terre.  Cosmos,  24,  543-5. 

133  1865    J.  H.  Pratt.       A  treatise  on  attractions,  Laplace's  functions, 

and  the  figure  of  the  earth.  3d.  Edn.  Cambridge 
and  London.  8V0. 

134  1865    H.  Scheffler.     Ueber  die  mittlere  Dichtigkeit  der  Erde.     Zeit. 

fur  Math.  it.  Phys.,  1O,  224-7. 

135  1869     A.  Schell.          Ueber  die  Bestimmung  der  mittleren  Dichtigkeit 

der  Erde.     Gottingen.     4to. 

136  1872    F.  Folie.  Sur  le  calcul  de  la  densite  moyenne  de  la  terre, 

d'apres  les  observations  d'Airy.  Bull.  Acad.  Roy. 
Belg.,  [2],  33,  369-372  and  389-409. 

137  1873    A.  Cornu  et  J.  B.  Bailie.    Determination   nouvelle  de  la  con- 

stante  de  1'attraction  et  de  la  densite  moyenuetle 
la  terre.  Comp.  Rend.,  76,  954-8. 


THE    LAWS    OF    GRAVITATION 


138*  1873    — 


139    1873 


145 
146 


1878 
1879 


(Notice  of  137).  Bull.  Ilebd.  de  I'Assoc.  Sclent,  de 
France.,  [1J,  12,  70. 

A.  Cornu  and  J.  B.  Bailie.  Mutual  determination  of  the  con- 
stant of  attraction,  and  of  the  menu  density  of 
the  earth.  Chemical  New*,  27,  211. 

140  1873    I.  Todhunter.  A  history  of  the  mathematical  theories  of  at- 

traction and  i he  figure  of  the  earth,  from  the 
time  of  Newton  to  that  of  Laplace.  2vol.  Lou- 
don.  8V0. 

141  1878     A.  Cornu  et  J.  B.  Bailie.     Elude  de  la  resistance  de  Fair  dans 

la  balance  de  torsion.     Comp.  Rend.,  86,  571-4. 

142  1878    A.  Cornu  et  J.  B.  Bailie.     Sur  la  mesure  de  la  densite  moyenne 

de  la  terre.     Comp.  Rend.,  86,  699-702. 

143  1878    A.  Cornu  et  J.  B.  Bailie.  Influence  des  termes  proportioned  an 

carre  des  ecarts,  dans  le  mouvemenl  oscillatoire 
dc  la  balance  de  torsion.  Comp.  Mend.,  86, 
1001-4. 

144  1878     Ph.  von  Jolly.  Die  Anwendung  der  Waage  auf  Probleme  der 

Gravitation.    Parti.     Abh.  Bay.  Akad.  Wiss.cl. 

2,  13,  Abth.  1,  157-176. 

Ph.  von  Jolly.  (Same  title  as  144).     Wied.  Ann.,  5,  112-34. 
J.  H.  Poynting.  On  a  method  of  using  the  balance  with  great 

delicacy,  and  on  its  employment  to  determine 

the  mean  density  of  the  earth.     Proc.  Roy.  Soc. 

Lond.,  28,  2-35. 
1880    H.  A.  E.  A.  Faye.     Sur  les  variations   seculaires  de  la  figure 

mathematique  de  la  terre.     Comp.  Rend.,  9O, 

1185-91. 

147  1880     H.  A.  E.  A.  Faye.    Sur  la  reduction  des  observations  du  pen- 

dule  an  niveau  de  la  mer.  Comp.  Rend.,  9O, 
1443-6. 

148  1880-4    F.  R.  Helmert.      Die   mathematischen    und    physikalischen 

Theorieen  der  hohercn  Geodasie.     2vol.    Leip- 
zig.    8V0. 
148|  1880-5    O.  Zanotti-Bianco.     II  problema  meccanico  della  figuradella 

terra.     2  parts.     Firenze  Torino-Roma.     8V0. 
A.  R.  Clarke.   Geodesy.     Oxford.     8V0. 
O.  Knopf.    Ueber  die  Methoden  zur  Bestimmung  der  mittleren 

Dichtigkeit  der  Erde.     Jena. 

T.  C.  Mendenhall.  Determination  of  the  acceleration  due  to  the 
force  of  gravity,  at  Tokio,  Japan.  Am.  Journ. 
Sc.,  [3],  2O,  124-32. 

T.  C.  Mendenhall.  On  a  determination  of  the  force  of  gravity 
at  the  summit  of  Fujiyama,  Japan.  Am.  Journ. 
Sc.,  [3],  21,  99-103. 

Sulla  diminuzione  della  gravita  coll'altezza. 
Atti  Accad.  Lincei.  Mem.  d.  sc.,  [3].  9,  103-17. 

153    1881     Ph.  von  Jolly.  (Same  title  as  144).     Part  2.     Abh.  Bay.  Akad. 
Wmcl.  2.  14,  Abth.  2,  3-26. 
153 


149  1880 
149^*  1880 

150  1880 


151  1881 


152    1881     F.  Keller. 


MEMOIRS    ON 

154  1881     Ph.  von  Jolly.  (Same  title  as  153).     Wied.  Ann.,  14,  331-55. 
154|  1882    J.  G.  Wallentin.     Ueber  die   Methoden    zur  Bestimmung  der 

mittleren  Dichte  der  Erde  und  eine  neue  dies- 
bezUgliche  Anwendung  der  Wage.  Humboldt, 
1,  212-7. 

155  1882    R.  von  Sterneck.     Untersucbungen  ilber  die   Scbwere  im  In- 

nern  der  Erde.  Mitth.  Mil.-Oeog.  Inst.  Wien,  2, 
77-120. 

156  1883    R.  von  Sterneck.  Wiederbohmg  der  Untersuchungen  tlber  die 

Schwere  im  Innern  der  Erde.  Mitth.  Mil.-Oeog. 
Inst.  Wien,  3,  59-94. 

157  1883    J.  B.  Bailie.      Sur  la  resistance  de  1'air  dans  les  mouvements 

oscillatoires  tres  lents.   Comp.  Rend.,  96, 1493-5. 

158  1884    R.  von  Sterneck.     Untersucbungen  tlber  die  Scbwere  auf  der 

Erde.     Mitth.  Mil.-Oeog.  Inst.  Wien,  4,  89-155. 

159  1884    A.  Kdnigand  F.  Richarz.    Eine  neue  Metbode  zur  Bestimmung 

der  Gravitationsconstante.  Sitzungsb.  Akad. 
Wiss.  Berlin,  1203-5. 

160  1885    A.  Kouig  and  P.  Ricbarz.     (Same  title  as  159).     Wied.  Ann., 

24,  664-8. 

161  1885    A.M.Mayer.    Methods  of  determining  the  density  of  tbe  earth. 

Nature,  31,  408-9. 

162  1885    A.  Konig  and  F.  Ricbarz.     Remarks  on  our  method  of  deter- 

mining tbe  mean  density  of  the  earth.  Nature, 
31,484. 

1J33  1885  J.  Wilsing.  Ueber  die  Anwendung  des  Pendels  zur  Bestim- 
mung der  mittleren  Dicbtigkeit  der  Erde.  Sitz- 
ungsb.  Akad.  Wiss.  Berlin,  Hbbd.  1,  13-15. 

164  1885     R.  von  Sterneck.     Fortsetzung  der  Uutersuchungen  ilber  die 

Sohwere  auf  der  Erde.  Mitth.  Mil.-Geog.  Inst. 
Wien,  5,  77-105. 

165  1886    R.  von  Sterneck.     (Same  title  as  155).     Mitth.  Mil.-Geog.  Inst. 

Wien,  6,  97-119. 

166  1886     W.  M.  Hicks.    On    some    irregulariiies  in  tbe    values  of  the 

mean  density  of  the  earth,  as  determined  by 
Baily.  Proc.  Cam.  Phil.  8oc.,  5,  pt.  2,  156-61. 

167  1886    F.  Keller.          Sul  metodo  di  Jolly  per  la  determinazione  della 

densita  media  della  terra.  Atti  Accad.  Lincei. 
Rend.,  [4],  2,  145-9. 

168  1887    F.  Keller.          Sulla  deviazione  del  filo  a  piombo  prodotta  dal 

prosciugamento  del  Lago  di  Fucino.  Atti  Accad. 
Lincei.  Rend.,  [4],  3,  493-501. 

169  1887    J.  Wilsing.        Mittheilung  tlber  die  Resultale  von  Pendelbeo- 

bachtungen  zur  Bestimmung  der  mittleren  Dich- 
tigkeit  der  Erde.  Sitzungsb.  Akad.  Wiss.  Berlin, 
Hbbd.  1,  327-34. 

170  1887    J.  Wilsing.        Bestimmung  der  mittleren  Dicbtigkeit  der  Erde 

mil  Hulfe  eines  Pendelapparates.     Publ.  Astro- 
phys.  Obs.  Potsdam,  6,  Stuck  2,  35-127. 
154 


THE    LAWS    OF    GRAVITATION 


171  1888  J.  H.  Gore.  Determination  of  the  mean  density  of  the  earth 
by  means  of  a  pendulum  principle,  by  J.  Wil- 
sing,  translated  and  condensed.  Smithsonian 
Rep.  1888.  635-46. 

(Same  title  as  170).  Publ.  Astrophys.  Obs.  Pots- 
dam, 6,  Stuck  3,  133-91. 

Ueber  einen  neuen  Apparat  zur  Bestimmung 
der  Erddichte.  Zeit.  fur  List.  -  Kunde.,  9, 
354-5. 

A  bibliography  of  geodesy.  Washington.  4to. 
App.  to  U.  S.  Coast  and  Geod.  Surv.  Rep.  for 
1887. 

On  the  Cavendish  Experiment.     Proc.  Roy.  Soc. 
Lond.,  46,  253-68. 
C.  V.  Boys.     (Same  title  as  175).     Nature,  41,  155-9. 


172  1889    J.  Wilsing. 

173  1889     W.  Laska. 


174    1889    J.  H.  Gore. 


175    1889    C.  V.  Boys. 


176 
177 


185 
186 


1889-90 
1889-90 


J.  Joly. 


178    1889-91    - 


179    1890    Thiesen 


180    1891 


181  1893  A.  Berget. 


182  1893  Gouy. 


183  1893 


184  1894 


1894 
1894 


187  1894 


188  1894 


(Report   of   meeting   of    Univ.   Exptl.    Assoc. 

Dublin).     Nature,  41,  256. 

Collection  de  memoires  relatifs  a  la  physique, 

publics  par  la  Societe  Francaise  de  Physique, 

4  and  5.     Paris.     8V0. 

Determination  de  la  variation  de  la  pesanteur 

avec  la  hauteur.   Trav.  et  Mem.  du  Bur.  Internat. 

des  Poids  et  Mes.,  7,  3-32. 
J.  H.  Poynting.     On  a  determination  of  the  mean  density  of 

the  earth  and  the  gravitation  constant  by  means 

of  the  common  balance.     Phil.   Trans.  Lond., 

[A],  182,565-656. 

Determination  experimentale  de  la  constantede 

Tattractiou  universelle,  ainsi  que  de  la  masse  et 

de  la  densite  de  la  terre.     Gomp.  Rend.,  116, 

1501-3. 

Sur  la  realisation  des  temperatures  constantes. 

Comp.  Rend.,  117,  96-7. 
F.  Richarz  und  O.  Kri.uar-Menzel.     Die  Abnahme  der  Schwere 

mil  der  Mohe  beslimmt  durch  Wagungen.    Sitz- 

ungsb.  Akad.   Wiss.  Berlin,  163-83. 
F.  Richarz  und  O.  Krigar-Menzel.     (Same  title  as  183).      Wied. 

Ann.,  51,  559-83. 

J.  H.  Poynting.     The  mean  density  of  the  earth.    London.  8V0. 
J.  H.  Poynting.     A  histciy  of  the  methods  of  weighing  the 

earth.     Proc.  Birmingham  Nat.  Hist,  and  Phil. 

Soc.,  9,  1-23. 

Die  Methoden   zur  Bestimmung  der   mittleren 

Dichte  der  Erde.    Wiss.  Beilage  zum  Jahresb.  des 

Gym.  zu  Weissenburg  i.  Elsass. 

On  the  Newtonian  constant  of  gravitation.  Proc. 

Roy.  Soc.  Lond.,  56,  131-2. 

(Same  title  as  187).     Nature,  5O,  330-4,  366-8, 

417-9  and  571. 
155 


186|  1894  G.  Fresdorf. 


C.  V.  Boys. 
C.  V.  Boys. 


MEMOIRS    ON    THE    LAWS    OF    GRAVITATION 

189  1895     C.  V.  Boys.      (Same  title  as  187).      Phil.  Trans.  Lond.,  [A] 

186,  1-72. 

190  1895    A.S.Mackenzie.    On  the  attractions  of  crystalline  and  isotropic 

masses  at  small  distances.  Phy.  Rev.,  2,  321- 
43. 

191  1896    F.  Richarz  und  O.  Krigar-Menzel.     Gravitationsconstante  uud 

mittlere  Dichtigkeit  der  Ertle,  bestimml  durch 
Wagungen.  Sitzungsb.  Akad.  Wiss.  Berlin, 
1305-18. 

192  1896    R.  von  Eotvos.     Untersuclmngen  liber  Gravitation   und  Erd- 

magmetismus.     JVied.  Ann.,  59,  354-400. 

193  1896    C.  Brauu.          Die  Gravitationsconstante,  die  Masse  und  mitt- 

lere Dichte  der  Erde  nach  einer  neuen  experi- 
mentellen  Bestimmung.  Denkschr.  Akad.  Wiss. 
Wien.  math.-nalurw.  cl.,  64,  187-258c. 

Iffi0-1  QQA -7    —  The  gravitation  constant  and  the  mean  density 

of  the  earth.     Nature,  55,  296. 

195  1897    P.  Drude.          Ueber  Fernewirkungen.     Wied.  Ann.,  62,  i.- 

xlix. 

196  1897    L.  W.  Austin  and  C.  B.  Thwing.     An  experimental  research  on 

gravitational  permeability.  Phy.  Rev.,  5,  294- 
300. 

196|  1897-9  S.  Gunthcr.  Handbuch  der  Geophysik.  2  vol.  Stuttgart. 
8V0. 

197  1897      J.  H.  P(oynting).     A  new  determination    of   ihe   gravitation 

constant  and  the  mean  density  of  the  earth. 
Nature,  56,  127-8. 

198  1898    F.  Richarz  und  O.  Krigar-Menzel.     Bestimmung  der  Gravita- 

tionsconslante  und  mittlercn  Dichtigkeit  der 
Erde  durch  Wauungen.  Anhang  Abh.  Akad. 
Wiss.  Berlin,  1-196. 

199  1898    F.  Richarz  und  O.  Krigar-Mcnzel.     (Same  title  as  191).     Wied. 

Ann.,  66,  177-193. 

^200  1899  J.  H.  Poynting  and  P.  L.  Gray.  An  experiment  in  search  of 
a  directive  action  of  one  quartz  crystal  on  an- 
other. Phil.  Trans.  Lond.,  [A],  192,  245-56. 


156 


INDEX 


Aehard,  49. 

Airy.  5,  106,  113.  118,  119,  121-124, 
128-130;  Theory  of  Cavendish  Ex- 
periment, 106,  118;  Dolcoath  Ex- 
periments, 113;  Harton  Experi- 
ments, 121. 

Arthur's  Seat,  118,  123,  124. 

Attraction,  Newton's  Theorems  on, 
9  ;  Newton's  Error  in  Calculation 
of,  16,  17  ;  Primitive,  27  ;  Of  a 
Plntenu,  29-32  ;  Of  a  Spherical 
Segment,  Calculated  by  Newton, 
17;  by  Carlini,  Schmidt  and  Giu- 
lio,  111,  112  ;  Shown  by  Deflection 
of  Plumb-line,  33-43  ;  Of  Chirnbo- 
razo,  34,  39  ;  Of  Schehallien,  43, 
53-56  ;  Due  to  Tides,  44,  134  ;  Of 
any  Hill,  Calculated  by  II niton, 
54*;  Of  the  Great  Pyramid,  55; 
Local,  56,  122-124,  126,  134,  135, 
141  ;  Of  Mount  Mimet,  56  ;  Of 
Mass  Beneath  Earth's  Surface,  56, 

122,  123;    Of  Arthur's  Seat,  118, 

123,  124 ;    Of  Evaux,  118  ;   Of   a 
Cone,  428 ;   Of  an  Infinite  Plane, 
135. 

Austin  and  Thwing,  142. 

B 

Babinet,  106. 

Bacon,  1,  2,  5.  49,  113 

Baily,  100,  105,  106,  115-120,  125, 
131-133,  137  ;  Cavendish  Experi- 
ment Criticized  by,  105  ;  Error  of, 
Pointed  out  by  Cornu  and  Bailie, 
119  ;  Anomalies  in  Results  of,  and 
their  Explanations,  118,  119. 

Balance,  Experiments  with  Beam, 
2-5,  48,  49,  125,  132.  140  ;  Experi- 
ments with  Torsion,  59-105,  114- 
121,  124,  135,  137-139.  142;  Mich- 
ell  Devised  Torsion,  60;  Experi- 
ments with  Pendulum,  131,  132. 


Baucrnfeind,  124. 

Beaumont,  116. 

Benret,  124,  134,  135. 

Bertier,  47-49. 

Boscovitch,  134. 

Bougner,  5,  21,  23-25,  27,  32,  33,  36, 
39-44,  47,  53,  56,  130,  134;  On 
Tides,  44,  134  ;  Life  of,  44  ;  First 
to  Take  Account  of  Buoyancy  of 
Air,  26. 

Boyle,  4. 

Boys,  106,  135-137,  139,  142. 

Brandes,  105,  106,  120;  Theory  of 
Cavendish  Experiment,  106;  The- 
ory of  Oscillation  Method,  105, 
106,  120. 

Braun,  106,  138,  139. 


Carlini.  111-113,  128. 

Cavendish,  54,  55,  59.  90,  91,  98,  100, 
105-107,  114-116,118,  119, 125, 135, 
136,  139  ;  Error  in  Calculation  of, 
100,  105  ;  Life  of,  107. 

Chimborazo,  22,  34.  39-41,  43. 

Clarke,  56,  118,  123,  124. 

Condamine,  de  la,  21,  28,  32,  36,  39- 
41,  43,  44  ;  Pendulum  Experiments 
of,  28  ;  Method  of,  for  Doubling 
Deflection  of  Plumb-line,  36. 

Cornu  and  Bailie,  66,  106,  115,  119, 
124,  125,  131,  135. 

Cotte,  48. 

Cotton,  4. 

Coulomb  Balance,  First  Proposed  by 
Michell,  60. 

Coultaud,  47,  48,  111. 


I) 


D'Alembert,  31.  47. 
Damping.    Method    of    Finding    A, 
138,  141,  142  ;  Effect  of,  139. 


157 


INDEX 


David,  47-49. 

Deflection,  of  Arm  of  Torsion  Bal- 
ance, How  Measured,  by  Caven- 
dish. 64,  98  ;  by  Reich,  116,  119  ; 
by  Baily,  117,  119,  132,  133;  by 
Braun.  '138  ;  Affects  the  Period, 
97 ;  Error  in  Daily's  Method  of 
Observing  119,  125  ;  Multiplied 
by  Poynting,  132,  133. 

Descartes,  2,  49  ;  Suggested  Method 
of  Measuring  Gravity,  2. 

Dimensions  of  Torsion  Balance,  Ef- 
fects of,  125,  135,  137,  138. 

Dolcoath,  113,  121. 

Dolomieu,  49. 

Drobisch,  113,  114. 

Drude,  142. 

E 

EotvOs,  106,  137,  138. 


Faye,  31,  124,  130,  139  ;  Compensa- 
tion Theory  of.  Correction  of  "  Dr. 
Young's  Rule,"  31,  124. 

Ferrel,  130. 

Flotation  Theory,  31,  124. 

Folie,  123. 

Forbes,  117,  118,  120. 

Forced  Vibrations,  J38, 141,  142. 

Fresdorf,  56.  112,  116,  123,  127,  128, 
131,  132,  135. 

Fujiyama,  127,  128. 


G 


Gilbert,  Dr.,  1,  5,49. 

Gilbert,  L.  W.,  105. 

Giulio,  112.      , 

Gore,  124,  132. 

Gosselin.  106. 

Gouy,  135. 

Gravimeter,  135. 

Gravitation,  Early  Conceptions  of, 
1,  49.  56  ;  Early  Experiments  on, 
by  Members  of  Royal  Society,  2-5  ; 
As  Explanation  of  Planetary  Mo- 
tion, by  Newton,  2,  10-19  :  Mag- 
netic Theory  of.  1, 4,  5. 12  ;  Hooke's 
Ideas  Concerning,  5,  6  ;  Compen- 
sator. 138 ;  Multiplicator,  138  ;  Per- 
meability, 142  ;  Velocity  of  Prop- 
agation of,  142. 

Gravity,  Proposed  Experiment  on, 
by  Bacon,  1  ;  by  Descartes,  2 ; 


Decrease  of,  with  Height,  27-33, 
47-49,  111-113,  126-128,  130,  137, 
140  ;  Law  of  Increase  of,  with 
Depth,  129,  130  ;  Increase  of,  with 
Temperature,  131  ;  Mathematical 
Discussion  of,  from  Potential  137 

Gray,  142. 

Giinther,  131,  141. 


H 


Harton  Colliery,  5,  121, 122 
Haughton.  122. 
Hearn,  118,  120. 
Helmert,  54,  56,  124,  127,  129. 
Hicks,  119,  131. 
Hooke,  2,  4,  5. 
Horizontalvariometer,  137. 
Humboldt,  56. 

Button,  54,  55,  90,  100,  105  106,  118, 
124. 

J 

Jacob,  56,  123. 

J;imes  and  Clarke,  56  118  123  124 
Jolly,  125, 126,  137,  140. 
Joly,  138,  141. 

K 

Keller,  124,  127,  135,  137,  140. 
Kepler,  1,  2,  49. 
Knopf,  113,  122. 
Konig,  140. 

Krigar-Menzel,  140,  141. 
Krummungsvariometer,  137,  138. 


Lalande,  48. 

Laska,  141. 

Law,  of  the  Distance,  2,  9  29  47 
101,  126,  142  ;  Of  the  Masses,  13, 
32;  Of  the  Material,  12,  142;  Of 
the  Medium,  142. 

Lesage,  2,  48,  49. 


M 


Mackenzie,  142. 

Magnetism.  Gilbert's  Explanation  of 
Gravitation  by,  1,4,  5  ;  Contrasted 
with  Gravitation,  12  ;  Tests  for 
Effects  of,  by  Cavendish,  67,  68 
75,  76  ;  by  Reich,  116,  120  ;  Sug- 
gested by  Hearn  to  Account  for 
Anomalies  in  Baily's  Results,  118- 
120. 


158 


INDEX 


Maskelyne,  17,  43,  53-56,  101,  106. 

Mayer,  140. 

Menabrea,  55,  66,  106. 

Mendenhall,  124,  127,  128. 

Mercier,  47,  48,  111. 

Michel],  59,  60,  61. 

Mine  Experiments,    1,   2,  4,  5,  49, 

113,  121,  128,  131. 
Montigny,  119. 
Muncke,  55,  105,  106. 

N 

Newton,  2,  6,  7,  9,  14-17,  19.  39,  43, 
47-49,  56,  107,  110,  124,  126,  141 ; 
Explanation  of  Planetary  Motions 
by,  2,  10  ;  Pendulum  Experiments 
of,  11,  15  ;  Guess  as  to  Value  of 
A  by,  14  ;  Errors  in  Calculations 
of,  16,  17  ;  Indicates  Methods  of 
Finding  A,  17;  Calculates  Attrac- 
tion of  a  Mountain,  17  ;  Life  of, 
19;  Attempts  to  Upset  Theory  of, 
47. 

P 

Pechmann,  124. 

Pendulum,  Experiment  with,  Pro- 
posed by  Bacon,  1  ;  by  Hooke,  5  ; 
Experiments  with,  by  Newion, 
10,  11,  15;  by  Bouguer,  24-33; 
by  Coultaud  and  Mercier,  47  ;  by 
Carlini,  111  ;  by  Airy,  113,  121  ; 
by  Mendenhall,  127  ;  by  Sterneck, 
128-131  ;  by  Laska,  141  ;  Correc- 
tion for  Buoyancy  of  Air  on,  First 
Used,  26  ;  Correction  for,  Due  to 
Resistance  of  Air,  27,  66  ;  Methods 
of  Comparing  One  with  Another, 
113,  121,  128-130;  Balance,  181. 

Peters,  55,  124. 

Playfair,  55,  102. 

Plumb-line,  Deflection  of,  Observed 
at  Chimborazo,  33-43  ;  at  Sche- 
hallien,  43,  53-56;  at  Arthur's 
Seat,  123;  at  Evaux,  118,  124; 
in  Tyrol,  124;  Deflection  of,  Cal- 
culated for  Chimborazo,  34;  how 
to  Observe,  35-39,  53 ;  by  Tides, 
44,  134,  135. 

Poisson,  31.  66 

Power,  2-5,  49. 

Poynting,  16, 36, 44, 106, 112, 116, 118, 
119, 123-125, 127, 128,  131-134, 142. 

Pratt,  124. 

Pringle,  56. 

Puissant,  118. 

Pyramid,  Attraction  of  the  Great,  55. 


R 

Reich,  91,  106,  114-121,  125,  138. 

Resistance  of  Air,  Discussed  by  Bou- 
guer, 27;  by  Cavendish,  65-67; 
by  Poisson,  Menabrea,  and  Coruu 
and  Bailie,  66,  106,  125. 

Richarz,  140,  141. 

Robison,  134. 

Roiffe,  48. 

Royal  Society,  Experiments  by 
Members  of,  2-6,  48. 

Rozier,  49. 


Sabine,  112. 

Saigey,  32,  43,  54,  112,  113,  118,  124; 

Correction  of  Peruvian  Pendulum 

Experiments  by,  32,  43. 
Schaar,  119. 
Scheffler,  123. 
Schehallien,  43,  53-55,  101, 112,  118, 

123. 

Schell,  56,  112,  116,  118,  123. 
Schmidt,  44,  55,  106,  112. 
Schubert,  124. 
Sheepshanks,  113. 
St.   Paul's   Cathedral,  Experiments 

at,  4,  5. 

Sterneck,  128-131,  137. 
Stokes,  122. 
Struve,  124,  134. 


Temperature,  Effects  of,  on  Torsion 
Balance,  Discussed  by  Cavendish, 
60,  76-80;  by  Reich,  114;  by 
Baily,  116,  117;  by  Hicks  and 
Poynting,  119  ;  by  Boys,  135,  136  ; 
by  Eotvos,  137  ;  by  Braun,  139  ; 
Change  of  A  with,  125,  131  ; 
Limit  of  Constancy  of,  135. 

Thiesen,  137. 

Thomson  and  Tail,  134. 

Tides,  Action  of,  on  Plumb-line,  44, 
134. 

Time  of  Vibration,  How  Found  by 
Cavendish,  64-67,  70  ;  by  Reich, 
115,  119  ;  by  Baily,  117  ;  by  Men- 
denhall, 127  ;  As  Affected  by  De- 
flection, 97  ;  As  Affected  by  Con- 
vection Currents,  80,  100,  134,  137  ; 
A  Found  From,  105,  106,  120, 
138,  139,  141. 

Todhunter,  16,  44,  56,  66. 


159 


INDEX 


U 


Ulloa,  25,  39,  40. 


Vacuum,  Experiment  Made  in,  138, 
141. 

W 

Wallentin,  127,  134. 

Westminster     Abbey,    Experiments 

Made  at,  2,  3,  5. 
W  he  well,  113. 


Wilsing,  131,  132. 

Y 

Young,  Rule  of,  31,  130. 

Z 

Zach,  36,  44,  54-56,  134  ;  Maskelyne 
Experiment  Calculated  by,  54  ; 
Finds  Attraction  of  Mount  Mi- 
met,  56. 

Zanotti-Bianco,  44,  54,  56,  106,  112, 
113,  123,  127. 


ADDENDUM 

Page  32.  [Faye  (146|)  has  calculated  the  diminution  in  the  attraction  ac- 
cording to  Ms  formula  (see  note  on  p.  31),  and  finds  it  to  be  the  ^^th  part, 
which  is  not  far  from  that  resulting  from  the  experiment.  His  calculation  can 
also  be  stated  in  the  following  way :  taking  no  account  of  the  attraction  of  the 
plateau,  tJie  observed  pendulum  lengtJis  reduced  to  sea-level  by  Saigey  are  at 


L'lsle  de  I'Inca   .     . 

Quito 

Difference 


990.935  mm. 
991.009    " 
.074    " 


which  difference  is  of  the  order  of  the  errors  of  the  observations.  See  this  vol- 
ume, p.  130,  Helmert  (148,  vol.  2,  chap.  3),  and  Zanotti-Bianco  (148%,  pt.  1, 
chap.  8,  andpt.  2,  p.  182).] 

160 


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